Explanation:
Okay, so I need to solve this problem about parallelograms bounded by lines and find the smallest possible value of a + b + c + d where a, b, c, d are positive integers. Let me try to break this down step by step.

First, the problem mentions two parallelograms. The first one is bounded by the lines y = ax + c, y = ax + d, y = bx + c, and y = bx + d. The area of this parallelogram is 18. The second parallelogram is bounded by y = ax + c, y = ax - d, y = bx + c, and y = bx - d, with area 72. We need to find the minimal a + b + c + d where all variables are positive integers.

Let me recall that the area of a parallelogram can be found using the base times height, but since these are parallelograms formed by lines with slopes a and b, maybe there's a formula involving the distance between the parallel lines and the angle between the lines?

Yes, I think the area of a parallelogram formed by two pairs of lines can be calculated by the product of the distances between the parallel lines and the sine of the angle between them. Let me verify that.

Suppose we have two pairs of parallel lines. The first pair is y = ax + c and y = ax + d. The distance between these two lines is |c - d| / sqrt(a² + 1). Similarly, the other pair is y = bx + c and y = bx + d, so the distance between them is |c - d| / sqrt(b² + 1). Wait, but in the first case, the lines are y = ax + c and y = ax + d, so the distance between them is |c - d| / sqrt(a² + 1). Similarly for the other pair. However, the area of the parallelogram would be the product of these two distances multiplied by the sine of the angle between the two lines?

Wait, actually, if you have two sets of parallel lines, the area of the parallelogram formed by their intersections is equal to the product of the distances between the lines divided by the sine of the angle between the lines? Wait, no, maybe it's the product of the distances multiplied by the sine of the angle between the lines?

Wait, let me think again. Let me recall that the area of a parallelogram is base times height. If I take one pair of lines as the base, then the distance between them is the height. But the other pair of lines would form the sides. Wait, maybe I need to think in terms of vectors.

Alternatively, maybe there's a formula for the area between two pairs of lines. Let me check.

Suppose we have two lines with slopes m1 and m2. The angle θ between them is given by tanθ = |(m2 - m1)/(1 + m1m2)|. Then, if we have two pairs of parallel lines, the distance between each pair is d1 and d2, then the area of the parallelogram would be d1 * d2 / sinθ? Or is it d1 * d2 * sinθ?

Wait, let me think of a simple case. Suppose we have two horizontal lines y = c and y = d, distance between them is |c - d|. Then another pair of vertical lines x = e and x = f, distance between them is |e - f|. The area of the parallelogram (which is actually a rectangle) is |c - d| * |e - f|. But in this case, the angle between the lines is 90 degrees, so sinθ = 1. So if I use the formula d1 * d2 / sinθ, that would be |c - d| * |e - f| / 1, which matches. If the angle is not 90 degrees, then maybe the formula is d1 * d2 / sinθ?

Wait, but in the case of a general parallelogram, if you have two pairs of lines with distances d1 and d2, and angle θ between them, then the area would be d1 * d2 / sinθ? Wait, no. Wait, if you have two adjacent sides of the parallelogram with lengths l and m, and angle θ between them, then the area is l * m * sinθ. But here, the distances between the lines are not the lengths of the sides. Wait, maybe I need to relate the distances between the lines to the sides of the parallelogram.

Alternatively, maybe the area can be calculated as the product of the distances between the lines divided by the sine of the angle between the lines? Wait, let me think again.

Let me take the first parallelogram. The lines y = ax + c and y = ax + d are parallel, as are y = bx + c and y = bx + d. The distance between the first pair is |c - d| / sqrt(a² + 1), and between the second pair is |c - d| / sqrt(b² + 1). The angle between the two lines is θ, where tanθ = |(b - a)/(1 + ab)|. Then, the area of the parallelogram would be the product of the distances between the lines divided by sinθ? Wait, if I think of the base as the distance between the first pair of lines, and the height as the distance between the second pair of lines divided by sinθ? Wait, maybe not.

Alternatively, if I imagine the parallelogram formed by these lines, then the sides of the parallelogram are along the directions of the lines y = ax + c and y = bx + c. The distance between the lines y = ax + c and y = ax + d is the height corresponding to the base along the direction of the lines y = bx + c. Wait, this is getting confusing. Maybe I need to use vectors.

Let me recall that the area of a parallelogram formed by two vectors is the magnitude of their cross product. So if I can find two vectors that form the sides of the parallelogram, then compute their cross product.

Alternatively, since the lines are given, maybe I can find the coordinates of the vertices of the parallelogram and then compute the area using coordinates.

Let me try that approach. Let me find the points of intersection of the lines.

First parallelogram: lines y = ax + c, y = ax + d, y = bx + c, y = bx + d.

So, the four lines are two pairs of parallel lines. The intersections will be:

Intersection of y = ax + c and y = bx + c: set ax + c = bx + c => (a - b)x = 0 => x = 0. Then y = a*0 + c = c. So point (0, c).

Similarly, intersection of y = ax + c and y = bx + d: ax + c = bx + d => (a - b)x = d - c => x = (d - c)/(a - b). Then y = a*(d - c)/(a - b) + c. Let me compute that:

y = [a(d - c) + c(a - b)] / (a - b) = [ad - ac + ac - bc]/(a - b) = [ad - bc]/(a - b). So the point is ((d - c)/(a - b), (ad - bc)/(a - b)).

Similarly, intersection of y = ax + d and y = bx + c: ax + d = bx + c => (a - b)x = c - d => x = (c - d)/(a - b) = -(d - c)/(a - b). Then y = a*(-(d - c)/(a - b)) + d = [-a(d - c) + d(a - b)]/(a - b) = [-ad + ac + ad - bd]/(a - b) = [ac - bd]/(a - b). So the point is (-(d - c)/(a - b), (ac - bd)/(a - b)).

Finally, intersection of y = ax + d and y = bx + d: ax + d = bx + d => (a - b)x = 0 => x = 0. Then y = a*0 + d = d. So point (0, d).

So the four vertices of the parallelogram are:

(0, c), ((d - c)/(a - b), (ad - bc)/(a - b)), (0, d), and (-(d - c)/(a - b), (ac - bd)/(a - b)).

Wait, but maybe I can compute the vectors between these points to find the area.

Alternatively, since it's a parallelogram, the area can be calculated by the base times height. But maybe using coordinates is more straightforward.

Alternatively, since the sides are between the lines, maybe we can compute the distance between the lines and the angle between them.

Wait, let me think again. The distance between the lines y = ax + c and y = ax + d is |c - d| / sqrt(a² + 1). Similarly, the distance between y = bx + c and y = bx + d is |c - d| / sqrt(b² + 1). The angle between the two lines is θ, where tanθ = |(b - a)/(1 + ab)|. Then, the area of the parallelogram is (distance between first pair) * (distance between second pair) / sinθ? Wait, if I think of the area as base * height, where base is the distance between the first pair of lines, and the height is the distance between the second pair of lines divided by sinθ? Wait, maybe not. Let me think.

Alternatively, if you have two vectors forming the sides of the parallelogram, then the area is the magnitude of the cross product. Let me try to find vectors.

Take two adjacent sides of the parallelogram. For example, from the point (0, c) to ((d - c)/(a - b), (ad - bc)/(a - b)). Let me compute this vector.

The vector is ((d - c)/(a - b) - 0, (ad - bc)/(a - b) - c) = ((d - c)/(a - b), (ad - bc - c(a - b))/(a - b)).

Simplify the y-component:

ad - bc - ac + bc = ad - ac = a(d - c). Therefore, the vector is ((d - c)/(a - b), a(d - c)/(a - b)) = (d - c)/(a - b) * (1, a).

Similarly, another vector from (0, c) to (-(d - c)/(a - b), (ac - bd)/(a - b)) is:

(-(d - c)/(a - b) - 0, (ac - bd)/(a - b) - c) = (-(d - c)/(a - b), (ac - bd - c(a - b))/(a - b)).

Simplify the y-component:

ac - bd - ac + bc = bc - bd = b(c - d) = -b(d - c). Therefore, the vector is (-(d - c)/(a - b), -b(d - c)/(a - b)) = -(d - c)/(a - b) * (1, b).

So the two vectors are (d - c)/(a - b)*(1, a) and -(d - c)/(a - b)*(1, b). Therefore, the area of the parallelogram is the magnitude of the cross product of these two vectors.

The cross product in 2D can be represented as the determinant:

| i    j    k |
| x1  y1   0 |
| x2  y2   0 |

Which is (x1 y2 - x2 y1) k. So the magnitude is |x1 y2 - x2 y1|.

So let me compute that.

First vector: ( (d - c)/(a - b), a(d - c)/(a - b) )

Second vector: ( -(d - c)/(a - b), -b(d - c)/(a - b) )

So x1 = (d - c)/(a - b), y1 = a(d - c)/(a - b)

x2 = -(d - c)/(a - b), y2 = -b(d - c)/(a - b)

Compute x1 y2 - x2 y1:

[(d - c)/(a - b)] * [-b(d - c)/(a - b)] - [ - (d - c)/(a - b) ] * [a(d - c)/(a - b) ]

Let me compute term by term:

First term: [(d - c)/(a - b)] * [-b(d - c)/(a - b)] = -b(d - c)^2 / (a - b)^2

Second term: - [ - (d - c)/(a - b) ] * [a(d - c)/(a - b) ] = + [ (d - c)/(a - b) * a(d - c)/(a - b) ] = a(d - c)^2 / (a - b)^2

Therefore, total is (-b + a)(d - c)^2 / (a - b)^2 = (a - b)(d - c)^2 / (a - b)^2 = (d - c)^2 / (a - b)

Therefore, the area is |(d - c)^2 / (a - b)|. Since a, b, c, d are positive integers, and the area is given as 18, so:

| (d - c)^2 / (a - b) | = 18.

But since the lines are given as y = ax + c and y = ax + d, and similarly for the other pair, we need to ensure that the denominators are not zero. Also, since a and b are slopes, they can be any real numbers, but in our case, they are positive integers. Also, since the lines are distinct, we need that c ≠ d and a ≠ b. Also, since the area is positive, the absolute value is considered. However, since the problem says that the area is 18, we can write:

(d - c)^2 / |a - b| = 18.

Similarly, for the second parallelogram, which is bounded by y = ax + c, y = ax - d, y = bx + c, and y = bx - d. Let me do the same analysis.

Following the same steps, the area would be similar. Let me check.

For the second parallelogram, the lines are y = ax + c, y = ax - d, y = bx + c, y = bx - d. So similar to the first case, but now the constants are c and -d instead of c and d. Let me compute the area.

Following the same method as before, the area would be |( (-d - c)^2 / (a - b) |? Wait, let me check.

Wait, in the first case, the difference between the constants was (d - c). In the second case, the constants are c and -d. So the difference between the constants for the first pair of lines (y = ax + c and y = ax - d) is c - (-d) = c + d. Similarly, for the second pair of lines (y = bx + c and y = bx - d), the difference is c - (-d) = c + d. Therefore, following the same formula as before, the area would be (c + d)^2 / |a - b| = 72.

Wait, let me verify this. Let me go through the same steps as before for the second parallelogram.

Let me find the vectors again. The lines are y = ax + c, y = ax - d, y = bx + c, y = bx - d.

Following similar steps as before, the points of intersection would be:

Intersection of y = ax + c and y = bx + c: (0, c) as before.

Intersection of y = ax + c and y = bx - d: ax + c = bx - d => (a - b)x = -d - c => x = -(c + d)/(a - b). Then y = a*(-(c + d)/(a - b)) + c = [-a(c + d) + c(a - b)]/(a - b) = [-ac - ad + ac - bc]/(a - b) = (-ad - bc)/(a - b) = -(ad + bc)/(a - b). So the point is (-(c + d)/(a - b), -(ad + bc)/(a - b)).

Similarly, intersection of y = ax - d and y = bx + c: ax - d = bx + c => (a - b)x = c + d => x = (c + d)/(a - b). Then y = a*(c + d)/(a - b) - d = [a(c + d) - d(a - b)]/(a - b) = [ac + ad - ad + bd]/(a - b) = (ac + bd)/(a - b). So the point is ((c + d)/(a - b), (ac + bd)/(a - b)).

Intersection of y = ax - d and y = bx - d: ax - d = bx - d => (a - b)x = 0 => x = 0. Then y = a*0 - d = -d. So point (0, -d).

So the four vertices are (0, c), (-(c + d)/(a - b), -(ad + bc)/(a - b)), (0, -d), and ((c + d)/(a - b), (ac + bd)/(a - b)).

Similarly, computing vectors from (0, c) to ((c + d)/(a - b), (ac + bd)/(a - b)):

x-component: (c + d)/(a - b) - 0 = (c + d)/(a - b)

y-component: (ac + bd)/(a - b) - c = [ac + bd - c(a - b)]/(a - b) = [ac + bd - ac + bc]/(a - b) = (bd + bc)/(a - b) = b(d + c)/(a - b)

So the vector is (c + d)/(a - b) * (1, b)

Similarly, the vector from (0, c) to (-(c + d)/(a - b), -(ad + bc)/(a - b)) is:

x-component: -(c + d)/(a - b) - 0 = -(c + d)/(a - b)

y-component: -(ad + bc)/(a - b) - c = [ -ad - bc - c(a - b) ] / (a - b) = [ -ad - bc - ac + bc ] / (a - b) = (-ad - ac)/ (a - b) = -a(d + c)/(a - b)

So the vector is -(c + d)/(a - b) * (1, a)

Therefore, the cross product of these two vectors is:

First vector: ( (c + d)/(a - b), b(c + d)/(a - b) )

Second vector: ( -(c + d)/(a - b), -a(c + d)/(a - b) )

So cross product determinant:

x1 y2 - x2 y1 = [ (c + d)/(a - b) * (-a(c + d)/(a - b)) ] - [ -(c + d)/(a - b) * b(c + d)/(a - b) ]

= [ -a(c + d)^2 / (a - b)^2 ] - [ -b(c + d)^2 / (a - b)^2 ]

= [ -a + b ] (c + d)^2 / (a - b)^2

= (b - a)(c + d)^2 / (a - b)^2

= -(a - b)(c + d)^2 / (a - b)^2

= - (c + d)^2 / (a - b)

Taking absolute value, the area is | - (c + d)^2 / (a - b) | = (c + d)^2 / |a - b|.

Therefore, the area of the second parallelogram is (c + d)^2 / |a - b| = 72.

So now, we have two equations:

From the first parallelogram: (d - c)^2 / |a - b| = 18.

From the second parallelogram: (c + d)^2 / |a - b| = 72.

Let me denote |a - b| as k. Since a and b are positive integers, and assuming a ≠ b (since otherwise the lines would be parallel and not form a parallelogram). Also, since the denominators are positive, we can write:

Let me assume that a > b, so that a - b is positive. Then |a - b| = a - b. So:

First equation: (d - c)^2 / (a - b) = 18.

Second equation: (c + d)^2 / (a - b) = 72.

Therefore, if I divide the second equation by the first equation, I get:

[(c + d)^2 / (a - b)] / [(d - c)^2 / (a - b)] = 72 / 18 => [(c + d)^2] / [(d - c)^2] = 4 => [(c + d)/(d - c)]^2 = 4 => (c + d)/(d - c) = ±2.

But since c and d are positive integers, and d - c must be positive (since otherwise the denominator would be negative or zero, but since we have (d - c)^2 in the first equation, which is positive, but if d - c is negative, then (d - c)^2 is same as (c - d)^2. However, in our case, since we have (d - c)^2 / (a - b) = 18, and a - b is positive, so (d - c)^2 must be positive, so d ≠ c. Also, since the lines are distinct, so d ≠ c. However, the sign of (d - c) depends on whether d > c or c > d.

But in the ratio [(c + d)/(d - c)]^2 = 4, taking square roots, we have (c + d)/|d - c| = 2. Because the square is 4, so the absolute value is 2. Therefore:

(c + d)/|d - c| = 2.

Therefore, c + d = 2 |d - c|.

Let me consider two cases:

Case 1: d > c. Then |d - c| = d - c. Therefore:

c + d = 2(d - c) => c + d = 2d - 2c => c + d - 2d + 2c = 0 => 3c - d = 0 => d = 3c.

Case 2: c > d. Then |d - c| = c - d. Therefore:

c + d = 2(c - d) => c + d = 2c - 2d => c + d - 2c + 2d = 0 => -c + 3d = 0 => c = 3d.

So in either case, we have that one of the variables is three times the other. So either d = 3c or c = 3d.

But since c and d are positive integers, both possibilities are valid. Let me note that.

So, let me consider both cases.

Case 1: d = 3c.

Then, substituting into the first equation:

(d - c)^2 / (a - b) = 18 => (3c - c)^2 / (a - b) = 18 => (2c)^2 / (a - b) = 18 => 4c² / (a - b) = 18 => 2c² / (a - b) = 9 => a - b = 2c² / 9.

Similarly, from the second equation:

(c + d)^2 / (a - b) = 72 => (c + 3c)^2 / (a - b) = 72 => (4c)^2 / (a - b) = 72 => 16c² / (a - b) = 72 => 2c² / (a - b) = 9, which is the same as the first equation. So that's consistent.

Therefore, in this case, a - b = 2c² / 9. Since a and b are positive integers, a - b must be a positive integer (since we assumed a > b). Therefore, 2c² must be divisible by 9. Therefore, 9 divides 2c². Since 2 and 9 are coprime, 9 divides c², so 3 divides c. Let me write c = 3k, where k is a positive integer. Then:

c = 3k, so d = 3c = 9k.

Then, a - b = 2*(3k)^2 / 9 = 2*9k² / 9 = 2k².

Therefore, a - b = 2k². Since a and b are positive integers, and a > b, we can write a = b + 2k².

Now, our variables are:

c = 3k,

d = 9k,

a = b + 2k².

We need to find positive integers a, b, c, d with these relations, and minimize a + b + c + d.

Similarly, let's check Case 2: c = 3d.

Then, substituting into the first equation:

(d - c)^2 / (a - b) = 18 => (d - 3d)^2 / (a - b) = 18 => (-2d)^2 / (a - b) = 18 => 4d² / (a - b) = 18 => 2d² / (a - b) = 9 => a - b = 2d² / 9.

Similarly, second equation:

(c + d)^2 / (a - b) = 72 => (3d + d)^2 / (a - b) = 72 => (4d)^2 / (a - b) = 72 => 16d² / (a - b) = 72 => 2d² / (a - b) = 9, same as before.

Therefore, similar to Case 1, a - b = 2d² / 9. Therefore, 2d² must be divisible by 9, so 9 divides 2d², so 3 divides d. Let me write d = 3m, where m is a positive integer. Then c = 3d = 9m.

Then, a - b = 2*(3m)^2 / 9 = 2*9m² / 9 = 2m². Therefore, a = b + 2m².

So variables:

d = 3m,

c = 9m,

a = b + 2m².

Again, need to minimize a + b + c + d = (b + 2m²) + b + 9m + 3m = 2b + 2m² + 12m.

Similarly, in Case 1, variables:

c = 3k,

d = 9k,

a = b + 2k²,

so a + b + c + d = (b + 2k²) + b + 3k + 9k = 2b + 2k² + 12k.

So both cases have similar expressions. Therefore, the minimal value of a + b + c + d would be the same in both cases, depending on k or m. However, since c and d are positive integers, and k and m are positive integers, we can consider both cases and find the minimal value.

But since the problem says that a, b, c, d are positive integers, and we need to find the minimal a + b + c + d, we can consider both cases and find the minimal possible value.

But perhaps there's a way to combine both cases. However, since in Case 1, c = 3k and d = 9k, and in Case 2, c = 9m and d = 3m, so they are symmetric. Therefore, the minimal value would be the same in both cases, so we can just consider one case and find the minimal value.

So let me focus on Case 1: c = 3k, d = 9k, a = b + 2k². Then, the total sum is 2b + 2k² + 12k. To minimize this, we need to choose k and b as small as possible.

But note that a and b must be positive integers. Since a = b + 2k², and b must be a positive integer, so b ≥ 1. Therefore, the minimal value of b is 1, and then a = 1 + 2k². Therefore, the total sum is 2*1 + 2k² + 12k = 2 + 2k² + 12k.

Similarly, for Case 2, the total sum is 2b + 2m² + 12m, with similar reasoning. Therefore, regardless of the case, the minimal sum would be achieved when k or m is as small as possible, and b is as small as possible.

But we need to check if there are any constraints on a and b. For example, in Case 1, a = b + 2k². Since a and b are positive integers, and b ≥ 1, then a is automatically positive. However, we need to ensure that the lines are distinct and that the denominators are non-zero. But since a ≠ b (as a - b = 2k² > 0), and c and d are positive integers, so all conditions are satisfied.

Therefore, to minimize the total sum, we need to minimize 2b + 2k² + 12k. Since b ≥ 1, the minimal value occurs when b = 1, and k is as small as possible. Let's check for k = 1:

Case 1, k = 1:

c = 3*1 = 3,

d = 9*1 = 9,

a = b + 2*(1)^2 = b + 2.

Total sum: 2b + 2*(1)^2 + 12*1 = 2b + 2 + 12 = 2b + 14. Since b ≥ 1, minimal when b = 1: 2*1 + 14 = 16. So total sum is 16 + a + ... Wait, no, wait:

Wait, the total sum is a + b + c + d = (b + 2k²) + b + c + d = 2b + 2k² + c + d. Wait, earlier I said 2b + 2k² + 12k. Wait, c = 3k, d = 9k, so c + d = 12k. Therefore, total sum is 2b + 2k² + 12k. So for k = 1, b = 1: 2*1 + 2*1 + 12*1 = 2 + 2 + 12 = 16. So total sum is 16. But let me check if this is valid.

Wait, a = b + 2k² = 1 + 2*1 = 3. So a = 3, b = 1, c = 3, d = 9. Let me check if this satisfies the original area conditions.

First parallelogram area: (d - c)^2 / (a - b) = (9 - 3)^2 / (3 - 1) = 36 / 2 = 18. Correct.

Second parallelogram area: (c + d)^2 / (a - b) = (3 + 9)^2 / 2 = 144 / 2 = 72. Correct. So this works. Therefore, the total sum is a + b + c + d = 3 + 1 + 3 + 9 = 16. Wait, but 3 + 1 + 3 + 9 is 16? 3 + 1 is 4, 3 + 9 is 12, total 16. Yes. So that's a possible value. But the problem says that a, b, c, d are positive integers. Here, a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid. But is this the minimal possible?

Wait, but maybe there are other cases where k is larger but b is smaller? Wait, no, since b is at least 1. However, maybe there are other cases where the ratio is different? Wait, but we considered both cases where d = 3c and c = 3d, and found that in both cases, the minimal sum is 16. However, let me check if there are other possibilities.

Wait, but maybe there are other solutions where the ratio (c + d)/|d - c| is 2, but with different relations? Wait, no, we considered both cases where d > c and c > d, leading to d = 3c or c = 3d. So those are the only possibilities.

But let me check if there are other possibilities where the ratio is -2? Wait, no, because we took absolute value. So the ratio is 2. Therefore, only those two cases.

But let me check if there are other solutions where the equations hold. For example, suppose that (c + d)^2 / (a - b) = 72 and (d - c)^2 / (a - b) = 18. Let me denote (d - c)^2 = 18 * (a - b) and (c + d)^2 = 72 * (a - b). Therefore, dividing the second equation by the first, we get [(c + d)^2]/[(d - c)^2] = 4, which leads to the same ratio as before. Therefore, the only possibilities are those two cases. Therefore, the minimal sum is 16? But let me check if there are smaller values.

Wait, let me check for k = 1, b = 1 gives total sum 16. Is there a way to get a smaller sum?

Suppose k = 1, b = 1: sum is 16.

What if k = 1, b = 1, but maybe there's another case where k is smaller? But k is a positive integer, so k cannot be smaller than 1.

Alternatively, maybe there are other solutions where the equations hold with different relations? For example, if (c + d)^2 / (a - b) = 72 and (d - c)^2 / (a - b) = 18, but with different relations between c and d?

Wait, but we already derived that the only possibilities are d = 3c or c = 3d. Therefore, there are no other possibilities. Therefore, the minimal sum is 16? But let me check if there are other solutions with different k and b.

For example, take k = 1, b = 1: sum 16.

If k = 1, b = 2: sum is 2*2 + 2*1 + 12*1 = 4 + 2 + 12 = 18, which is larger.

Similarly, for k = 2, b = 1: sum is 2*1 + 2*(2)^2 + 12*2 = 2 + 8 + 24 = 34, which is larger.

Alternatively, if we take Case 2, where c = 9m and d = 3m. Let me check for m = 1:

c = 9*1 = 9, d = 3*1 = 3. Then, a - b = 2*(3m)^2 / 9 = 2*(9)/9 = 2. Therefore, a = b + 2. Then total sum is a + b + c + d = (b + 2) + b + 9 + 3 = 2b + 14. Again, minimal when b = 1: 2 + 14 = 16. Which is same as before. So same result.

Therefore, regardless of case, the minimal sum is 16. However, let me check if there are other possibilities where the equations hold with different relations. For example, if the ratio [(c + d)/(d - c)] is 2, but with different values?

Wait, but we already considered that. So unless there are other solutions where [(c + d)/(d - c)] is 2, but with different relations. But no, because we derived that only d = 3c or c = 3d.

But let me check if there are other solutions where the equations hold. For example, suppose that (d - c)^2 / (a - b) = 18 and (c + d)^2 / (a - b) = 72. Let me denote x = d - c and y = c + d. Then, we have:

x² / k = 18,

y² / k = 72,

where k = a - b.

Therefore, y² / x² = 72 / 18 = 4 => y / x = 2 => y = 2x.

But y = c + d and x = d - c. Therefore:

c + d = 2(d - c) => c + d = 2d - 2c => 3c = d, which is Case 1.

Or, if we consider y = -2x, but since y and x are positive? Wait, x = d - c. If d < c, then x is negative, but since we have x², it's same as (c - d)^2. However, in the ratio y / x, if x is negative, then y / x = -2. But since y = c + d is positive, and x = d - c could be negative. However, since we have y² / x² = 4, so y / |x| = 2. Therefore, y = 2|x|. Therefore, if x = d - c is negative, then |x| = c - d, so y = 2(c - d). Therefore:

c + d = 2(c - d) => c + d = 2c - 2d => -c + 3d = 0 => c = 3d, which is Case 2.

Therefore, only those two cases. Therefore, there are no other solutions. Therefore, the minimal sum is 16. However, let me check if there are other solutions where k is not integer? Wait, no, because in our equations, we have that a - b = 2k², and since a and b are integers, then 2k² must be integer, which it is, since k is integer. Therefore, all solutions are covered in these cases.

But wait, the problem says that a, b, c, d are positive integers. In our case, with k = 1, we have a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid. However, let me check if there are smaller values. For example, if we take k = 1, but maybe different b?

Wait, if we take b = 1, then a = 3. If we take b = 0, but b must be positive integer, so b cannot be 0. Therefore, the minimal b is 1. Therefore, 16 is the minimal sum? But let me check if there are other possibilities where the equations hold with different relations. For example, if the ratio is different?

Wait, but we derived that [(c + d)/(d - c)]^2 = 4, so the ratio is 2. Therefore, no other possibilities. Therefore, 16 is the minimal sum. However, let me check if there are other solutions with different k and b where the total sum is less than 16.

For example, suppose k = 1, b = 1 gives sum 16. What if there is another solution where k is not integer? But no, since k is defined as c = 3k, so k must be integer. Therefore, no. Therefore, 16 is the minimal sum. However, let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of c and d?

Wait, for example, suppose that (d - c)^2 = 18 * k and (c + d)^2 = 72 * k. Therefore, (c + d)^2 = 4*(d - c)^2. Therefore, c + d = 2(d - c) or c + d = -2(d - c). Which leads to the same cases as before. Therefore, no other solutions.

But let me check with numbers. Suppose that c = 1, d = 3. Then, d - c = 2, c + d = 4. Then, 4^2 / (a - b) = 72 => 16 / (a - b) = 72 => a - b = 16 / 72 = 2/9. Not integer. Therefore, invalid. Similarly, if c = 2, d = 6. Then d - c = 4, c + d = 8. Then 8^2 / (a - b) = 72 => 64 / (a - b) = 72 => a - b = 64 / 72 = 8/9. Not integer. Similarly, c = 3, d = 9: d - c = 6, c + d = 12. Then 12² / (a - b) = 72 => 144 / (a - b) = 72 => a - b = 2. Which is valid. Which is our previous case. Therefore, this gives a - b = 2. Which is the case when k = 1, since c = 3k = 3*1 = 3, d = 9k = 9*1 = 9. Therefore, this is the same as before.

Similarly, if c = 1, d = 3, but then a - b = 2c² / 9 = 2*1 / 9 = 2/9, which is not integer. Therefore, invalid. Therefore, only when c is multiple of 3, then 2c² / 9 is integer. Therefore, c must be multiple of 3. Therefore, the minimal c is 3, leading to k = 1. Therefore, the minimal sum is 16.

But wait, the problem says that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid. However, let me check if there is another combination with smaller sum. For example, if we take k = 1, but different b? For example, if b = 1, sum is 16. If b = 2, sum is 2*2 + 2*1 + 12*1 = 4 + 2 + 12 = 18. Which is larger. If we take k = 1, b = 1, sum is 16. If we take k = 1, b = 1, and check if there are other variables. For example, a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid.

But let me check if there are other solutions where the equations hold with different relations. For example, suppose that (d - c)^2 / (a - b) = 18 and (c + d)^2 / (a - b) = 72, but with different values of c and d. For example, suppose that (d - c) = 3 and (c + d) = 6. Then, 6² / (a - b) = 72 => 36 / (a - b) = 72 => a - b = 0.5. Not integer. Not valid. If (d - c) = 6, (c + d) = 12, then 12² / (a - b) = 72 => 144 / (a - b) = 72 => a - b = 2. Which is our previous case. If (d - c) = 9, (c + d) = 18, then 18² / (a - b) = 72 => 324 / (a - b) = 72 => a - b = 324 / 72 = 4.5. Not integer. Not valid. If (d - c) = 12, (c + d) = 24, then 24² / (a - b) = 72 => 576 / (a - b) = 72 => a - b = 8. Then, (d - c) = 12, so d = c + 12. Also, c + d = 24 => c + c + 12 = 24 => 2c = 12 => c = 6, d = 18. Then, a - b = 8. Then, total sum is a + b + c + d = (b + 8) + b + 6 + 18 = 2b + 32. Minimal when b = 1: 2 + 32 = 34, which is larger than 16. Therefore, not better.

Alternatively, if (d - c) = 2, then (c + d) = 4. Then, 4² / (a - b) = 72 => 16 / (a - b) = 72 => a - b = 16 / 72 = 2/9. Not integer. Not valid.

Therefore, the only valid solutions are when (d - c) is multiple of 6, leading to c = 3k, d = 9k. Therefore, the minimal sum is 16. However, let me check if there are other possibilities where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of (d - c) and (c + d). For example, if (d - c) = 6, then (c + d) = 12, which gives us the case we already considered. If (d - c) = 3, then (c + d) = 6, but that leads to a - b = 2/9, which is invalid. Similarly, (d - c) = 9, (c + d) = 18, which gives a - b = 4.5, invalid. Therefore, only multiples where (d - c) is multiple of 6, leading to c = 3k, d = 9k.

Therefore, the minimal sum is 16. However, let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of a and b. For example, suppose that a - b is different. Wait, but we have that a - b is determined by the equations. For example, in the case where c = 3k, d = 9k, then a - b = 2k². Therefore, if k = 1, a - b = 2. If k = 2, a - b = 8, etc. Therefore, the minimal a - b is 2, which gives us the minimal sum.

But let me check if there are other solutions where a - b is different. For example, suppose that (d - c)^2 = 18 * (a - b) and (c + d)^2 = 72 * (a - b). Let me suppose that (d - c) = 6 and (c + d) = 12, which gives a - b = 2. Which is our case. Alternatively, if (d - c) = 12 and (c + d) = 24, then a - b = 144 / 72 = 2. Wait, no:

Wait, (c + d)^2 / (a - b) = 72. If (c + d) = 24, then 24² / (a - b) = 72 => 576 / (a - b) = 72 => a - b = 576 / 72 = 8. But (d - c) = 12, so (d - c)^2 / (a - b) = 144 / 8 = 18. Which is correct. Therefore, this is another solution. In this case, c + d = 24, d - c = 12. Therefore, solving:

c + d = 24,

d - c = 12.

Adding equations: 2d = 36 => d = 18, then c = 24 - 18 = 6. Therefore, c = 6, d = 18. Then, a - b = 8. Then, total sum is a + b + c + d = (b + 8) + b + 6 + 18 = 2b + 32. Minimal when b = 1: 2 + 32 = 34. Which is larger than 16. Therefore, not better.

Therefore, the minimal sum is indeed 16. However, let me check if there are other solutions with smaller sum. For example, if we take k = 1, but different values of a and b? Wait, no, because a = b + 2k². If k = 1, then a = b + 2. Therefore, if b = 1, a = 3. If b = 2, a = 4, etc. Therefore, the minimal sum occurs when b is minimal, which is 1. Therefore, 16 is the minimal sum.

But let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of c and d. For example, suppose that (d - c) = 6 and (c + d) = 12, which gives us c = 3, d = 9. Which is our case. Alternatively, if (d - c) = 6 and (c + d) = 12, but with different c and d? No, because solving c + d = 12 and d - c = 6 gives unique solution c = 3, d = 9. Therefore, no other solutions.

Alternatively, if (d - c) = 6 and (c + d) = 12, then same as above. Therefore, no other solutions. Therefore, the minimal sum is 16.

But wait, the problem says that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9. All positive integers. However, let me check if there are other solutions with smaller sum. For example, if we take k = 1, but with different b? No, since b is minimal 1. Therefore, 16 is the minimal sum.

But let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of a and b. For example, suppose that a - b = 1. Then, from the first equation: (d - c)^2 = 18 * 1 = 18. Therefore, d - c = sqrt(18) = 3√2, which is not integer. Therefore, invalid. Similarly, a - b = 2: (d - c)^2 = 36 => d - c = 6. Then, (c + d)^2 = 72 * 2 = 144 => c + d = 12. Which gives c = 3, d = 9. Which is our case. If a - b = 3: (d - c)^2 = 18 * 3 = 54. Not a perfect square. Similarly, a - b = 4: (d - c)^2 = 72. Not a perfect square. a - b = 6: (d - c)^2 = 108. Not a perfect square. a - b = 8: (d - c)^2 = 144 => d - c = 12. Then, (c + d)^2 = 72 * 8 = 576 => c + d = 24. Which gives c = 6, d = 18. Which is another solution, but sum is higher. Therefore, the minimal a - b is 2, leading to the minimal sum of 16.

Therefore, the answer should be 16. However, let me check if there are other solutions with a - b = 2 and different c and d. For example, if a - b = 2, then (d - c)^2 = 18 * 2 = 36 => d - c = 6. And (c + d)^2 = 72 * 2 = 144 => c + d = 12. Therefore, solving:

c + d = 12,

d - c = 6.

Adding equations: 2d = 18 => d = 9, then c = 3. Therefore, unique solution. Therefore, only one solution for a - b = 2. Therefore, the minimal sum is 16.

But let me check if there are other solutions where a - b is different but the total sum is smaller. For example, if a - b = 2, sum is 16. If a - b = something else, but with smaller total sum? For example, if a - b = 2, and c = 3, d = 9, sum is 16. If there is another solution with a - b = 2, but different c and d? No, because c and d are uniquely determined by the equations. Therefore, no.

Therefore, the minimal possible value of a + b + c + d is 16. However, let me check the problem statement again to make sure I didn't miss anything.

Wait, the problem says that the parallelogram bounded by y = ax + c, y = ax + d, y = bx + c, y = bx + d has area 18. The other one has area 72. Given that a, b, c, d are positive integers. What is the smallest possible value of a + b + c + d?

Yes, and in our case, a = 3, b = 1, c = 3, d = 9. All positive integers. Sum is 3 + 1 + 3 + 9 = 16. However, let me check if there are other solutions with smaller sum. For example, if there is a case where c and d are smaller. For example, if c = 1, d = 3, but then (d - c)^2 / (a - b) = 4 / (a - b) = 18 => a - b = 4 / 18 = 2/9, which is not integer. Similarly, c = 2, d = 6: (6 - 2)^2 / (a - b) = 16 / (a - b) = 18 => a - b = 16 / 18 = 8/9, not integer. c = 3, d = 9: gives a - b = 2, which is valid. Therefore, no smaller c and d possible. Therefore, 16 is the minimal sum.

But wait, the problem says that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9. All positive integers. However, let me check if there is another solution where a and b are swapped? For example, if a = 1, b = 3. But then a - b = -2, but we took absolute value earlier. However, in our equations, we assumed a > b, but if a < b, then |a - b| = b - a. However, in our equations, we had:

From the first parallelogram: (d - c)^2 / |a - b| = 18.

From the second: (c + d)^2 / |a - b| = 72.

Therefore, even if a < b, the equations are same. Therefore, if we take a = 1, b = 3, then |a - b| = 2, same as before. Then, same equations. Therefore, c = 3, d = 9. Then, a + b + c + d = 1 + 3 + 3 + 9 = 16. Same sum. Therefore, same result.

But in this case, a = 1, b = 3, c = 3, d = 9. All positive integers. Therefore, same sum. Therefore, regardless of which is larger between a and b, the sum remains same.

But let me check if there are other solutions with different a and b. For example, if a - b = 2, then a = b + 2. If we take b = 2, then a = 4. Then, c = 3, d = 9. Then, sum is 4 + 2 + 3 + 9 = 18, which is larger than 16. Therefore, not better.

Therefore, the minimal sum is indeed 16. However, let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of c and d. For example, if c = 6, d = 18, then (d - c)^2 / (a - b) = 144 / (a - b) = 18 => a - b = 8. Then, (c + d)^2 / (a - b) = 24² / 8 = 576 / 8 = 72. Correct. Therefore, this is another solution. Then, a - b = 8. If we take b = 1, then a = 9. Then, sum is 9 + 1 + 6 + 18 = 34. Which is larger than 16. Therefore, not better.

Therefore, the minimal sum is 16. However, let me check if there are other solutions with smaller sum. For example, if we take k = 1, but different values of a and b? No, because a = b + 2. Therefore, minimal when b = 1. Therefore, 16 is the minimal.

But wait, the problem says that a, b, c, d are positive integers. In our case, c = 3 and d = 9. However, what if there are other solutions where c and d are smaller? For example, if c = 3, d = 9, but with different a and b? No, because a - b is fixed once c and d are fixed. Therefore, no.

Alternatively, if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of (d - c) and (c + d). For example, if (d - c) = 6 and (c + d) = 12, which gives us the case we already considered. Therefore, no other solutions.

Therefore, I think the minimal sum is 16. However, I need to check if there are any constraints that I missed. For example, the problem says that the lines are y = ax + c, y = ax + d, etc. Are there any restrictions on the slopes? For example, if a and b are positive integers, and the lines are distinct. Since a and b are positive integers, and we have a ≠ b (since a - b is non-zero). Therefore, all conditions are satisfied.

Therefore, I think the answer is 16. However, let me check once again with the given values:

a = 3, b = 1, c = 3, d = 9.

First parallelogram: lines y = 3x + 3, y = 3x + 9, y = 1x + 3, y = 1x + 9.

Area is 18. Let me compute it using the formula we derived: (d - c)^2 / |a - b| = (9 - 3)^2 / |3 - 1| = 36 / 2 = 18. Correct.

Second parallelogram: lines y = 3x + 3, y = 3x - 9, y = 1x + 3, y = 1x - 9.

Area is (c + d)^2 / |a - b| = (3 + 9)^2 / 2 = 144 / 2 = 72. Correct.

Therefore, all conditions are satisfied. Therefore, the minimal sum is 16. However, the problem says "the smallest possible value of a + b + c + d". But let me check if there is a case where a, b, c, d are smaller. For example, if there is a case where a = 1, b = 3, c = 3, d = 9, which gives the same sum. Or if there is another combination with smaller numbers.

Wait, let me think. Suppose that instead of c = 3, d = 9, we have c = 1, d = 3, but then as before, a - b = 2/9, which is invalid. Therefore, no. Therefore, 16 is the minimal.

But wait, the problem says that a, b, c, d are positive integers. In our case, c = 3 and d = 9. However, what if there is a case where c and d are smaller? For example, if c = 3, d = 9, but with different a and b? No, because a - b is fixed. Therefore, no.

Alternatively, if there is a case where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of (d - c) and (c + d). For example, if (d - c) = 6 and (c + d) = 12, which gives us the case we already considered. Therefore, no.

Therefore, I think the answer is 16. However, I need to check if there is a possible mistake in my calculation. For example, when I calculated the area of the parallelogram, I used the formula (d - c)^2 / |a - b|. Let me verify this with coordinates.

Take the first parallelogram with a = 3, b = 1, c = 3, d = 9. The four vertices are:

(0, 3), ((d - c)/(a - b), (ad - bc)/(a - b)) = (6 / 2, (27 - 3)/2) = (3, 12), (0, 9), and (-3, (ac - bd)/(a - b)) = (-3, (9 - 27)/2) = (-3, -9). Wait, let me compute the coordinates again.

Wait, earlier, we had for the first parallelogram:

Vertices are (0, c) = (0, 3), ((d - c)/(a - b), (ad - bc)/(a - b)) = ( (9 - 3)/(3 - 1), (3*9 - 1*3)/(3 - 1) ) = (6/2, (27 - 3)/2) = (3, 24/2) = (3, 12). Then (0, d) = (0, 9), and the fourth vertex is (-(d - c)/(a - b), (ac - bd)/(a - b)) = (-3, (3*3 - 1*9)/2) = (-3, (9 - 9)/2) = (-3, 0). Wait, that's different from what I wrote earlier. Wait, earlier I thought the fourth vertex was (-3, -9), but that seems incorrect. Let me recalculate.

Wait, the fourth vertex is intersection of y = ax + d and y = bx + c. Which we calculated earlier as ((c - d)/(a - b), (ac - bd)/(a - b)). Wait, no:

Wait, earlier, for the first parallelogram, the intersection of y = ax + d and y = bx + c is:

ax + d = bx + c => (a - b)x = c - d => x = (c - d)/(a - b) = -(d - c)/(a - b). Then y = a*x + d = a*(-(d - c)/(a - b)) + d = [-a(d - c) + d(a - b)] / (a - b) = [-ad + ac + ad - bd]/(a - b) = [ac - bd]/(a - b). Therefore, the point is (-(d - c)/(a - b), (ac - bd)/(a - b)).

In our case, a = 3, b = 1, c = 3, d = 9. Therefore:

x = -(9 - 3)/(3 - 1) = -6/2 = -3.

y = (3*3 - 1*9)/(3 - 1) = (9 - 9)/2 = 0/2 = 0. Therefore, the fourth vertex is (-3, 0). Therefore, the four vertices are (0, 3), (3, 12), (0, 9), (-3, 0). Wait, but how does this form a parallelogram?

Let me plot these points:

(0, 3), (3, 12), (0, 9), (-3, 0). Wait, connecting these points:

From (0, 3) to (3, 12): this is one side.

From (0, 3) to (-3, 0): another side.

From (3, 12) to (0, 9): another side.

From (0, 9) to (-3, 0): another side.

Wait, but to compute the area, maybe using coordinates. Let me use the shoelace formula.

List the coordinates in order:

Let me take the points in order: (0, 3), (3, 12), (0, 9), (-3, 0), back to (0, 3).

Shoelace formula:

Area = 1/2 |sum_{i=1 to n} (x_i y_{i+1} - x_{i+1} y_i)|

Compute each term:

First term: x1 y2 - x2 y1 = 0*12 - 3*3 = 0 - 9 = -9

Second term: x2 y3 - x3 y2 = 3*9 - 0*12 = 27 - 0 = 27

Third term: x3 y4 - x4 y3 = 0*0 - (-3)*9 = 0 + 27 = 27

Fourth term: x4 y1 - x1 y4 = (-3)*3 - 0*0 = -9 - 0 = -9

Sum: -9 + 27 + 27 - 9 = 36

Area = 1/2 * |36| = 18. Correct. Therefore, the area is indeed 18. Similarly, for the second parallelogram, let me check.

Second parallelogram: lines y = 3x + 3, y = 3x - 9, y = 1x + 3, y = 1x - 9.

Vertices:

Intersection of y = 3x + 3 and y = x + 3: (0, 3).

Intersection of y = 3x + 3 and y = x - 9: 3x + 3 = x - 9 => 2x = -12 => x = -6. y = 3*(-6) + 3 = -18 + 3 = -15. So point (-6, -15).

Intersection of y = 3x - 9 and y = x + 3: 3x - 9 = x + 3 => 2x = 12 => x = 6. y = 6 + 3 = 9. So point (6, 9).

Intersection of y = 3x - 9 and y = x - 9: 3x - 9 = x - 9 => 2x = 0 => x = 0. y = -9. So point (0, -9).

Therefore, the four vertices are (0, 3), (-6, -15), (0, -9), (6, 9). Let me apply shoelace formula.

Order of points: (0, 3), (-6, -15), (0, -9), (6, 9), back to (0, 3).

Compute terms:

First term: x1 y2 - x2 y1 = 0*(-15) - (-6)*3 = 0 + 18 = 18

Second term: x2 y3 - x3 y2 = (-6)*(-9) - 0*(-15) = 54 - 0 = 54

Third term: x3 y4 - x4 y3 = 0*9 - 6*(-9) = 0 + 54 = 54

Fourth term: x4 y1 - x1 y4 = 6*3 - 0*9 = 18 - 0 = 18

Sum: 18 + 54 + 54 + 18 = 144

Area = 1/2 * |144| = 72. Correct. Therefore, the area is indeed 72. Therefore, the values are correct.

Therefore, the minimal sum is 16. However, I need to check if there is a case where the sum is smaller. For example, if there is a case where a, b, c, d are smaller. For example, if a = 2, b = 1, c = ?, d = ?. Let me check.

Suppose a = 2, b = 1. Then, |a - b| = 1. Then, from the first equation: (d - c)^2 / 1 = 18 => (d - c)^2 = 18. Not possible since 18 is not a perfect square. Therefore, invalid.

If a = 4, b = 2, then |a - b| = 2. Then, (d - c)^2 = 36 => d - c = 6. And (c + d)^2 = 72 * 2 = 144 => c + d = 12. Therefore, solving c + d = 12, d - c = 6 => d = 9, c = 3. Therefore, a = 4, b = 2, c = 3, d = 9. Sum is 4 + 2 + 3 + 9 = 18, which is larger than 16.

If a = 2, b = 1, no solution. If a = 1, b = 3, then |a - b| = 2. Then, same as before, c = 3, d = 9. Sum is 1 + 3 + 3 + 9 = 16. Same as before.

Therefore, no smaller sum. Therefore, the answer is 16.

But wait, the problem says "the smallest possible value of a + b + c + d". However, I need to check if there are other solutions with different a, b, c, d. For example, if a = 1, b = 3, c = 3, d = 9, sum is 16. If a = 1, b = 3, c = 3, d = 9, or a = 3, b = 1, c = 3, d = 9. Both give same sum. Are there other combinations?

For example, if a = 1, b = 3, c = 3, d = 9. Or if a = 1, b = 3, c = 3, d = 9. Or if a = 1, b = 3, c = 3, d = 9. No other combinations. Therefore, 16 is the minimal.

But let me check if there are solutions where c and d are swapped. For example, if c = 9, d = 3. Then, in the first case, (d - c)^2 = (3 - 9)^2 = 36, same as before. Then, (c + d)^2 = (9 + 3)^2 = 144. Therefore, same equations. Therefore, a - b = 2. Therefore, a = b + 2. Then, if b = 1, a = 3. Therefore, sum is 3 + 1 + 9 + 3 = 16. Same as before. Therefore, same result.

Therefore, regardless of the order of c and d, the sum remains same. Therefore, 16 is the minimal sum.

However, I need to check if there are other solutions where the equations hold with different values. For example, if a - b = 2, and c = 3, d = 9, or c = 6, d = 18, etc. But as we saw, the latter gives higher sum.

Therefore, I think the answer is 16. However, I need to check if there is a possible mistake in my initial assumption. For example, when I derived the area formula, I assumed that the area is (d - c)^2 / |a - b|. Let me verify this with another example.

Take a = 1, b = 2, c = 1, d = 3. Then, the first parallelogram area should be (3 - 1)^2 / |1 - 2| = 4 / 1 = 4. Let me compute it with coordinates.

Lines y = x + 1, y = x + 3, y = 2x + 1, y = 2x + 3.

Vertices:

Intersection of y = x + 1 and y = 2x + 1: x + 1 = 2x + 1 => x = 0, y = 1. Point (0, 1).

Intersection of y = x + 1 and y = 2x + 3: x + 1 = 2x + 3 => -x = 2 => x = -2, y = -2 + 1 = -1. Point (-2, -1).

Intersection of y = x + 3 and y = 2x + 1: x + 3 = 2x + 1 => -x = -2 => x = 2, y = 2 + 3 = 5. Point (2, 5).

Intersection of y = x + 3 and y = 2x + 3: x + 3 = 2x + 3 => x = 0, y = 3. Point (0, 3).

So vertices are (0, 1), (-2, -1), (0, 3), (2, 5). Applying shoelace formula:

Order: (0, 1), (-2, -1), (0, 3), (2, 5), back to (0, 1).

Compute terms:

First term: 0*(-1) - (-2)*1 = 0 + 2 = 2

Second term: -2*3 - 0*(-1) = -6 - 0 = -6

Third term: 0*5 - 2*3 = 0 - 6 = -6

Fourth term: 2*1 - 0*5 = 2 - 0 = 2

Sum: 2 - 6 - 6 + 2 = -8

Area = 1/2 * | -8 | = 4. Correct. Which matches the formula. Therefore, the formula is correct.

Therefore, my formula is correct. Therefore, the answer is 16.

But wait, the problem says "the smallest possible value of a + b + c + d". However, I need to check if there are other solutions with smaller sum. For example, if there is a case where a - b = 2, and c and d are smaller. But as we saw, c and d must satisfy c + d = 12 and d - c = 6, leading to c = 3, d = 9. Therefore, no smaller c and d possible. Therefore, 16 is indeed the minimal.

Therefore, I think the answer is 16. However, I need to check if the problem has any constraints that I missed. For example, the problem says that the lines are y = ax + c, y = ax + d, etc. Are there any restrictions on the lines? For example, they must be distinct, which they are since c ≠ d and a ≠ b. Also, the problem says that a, b, c, d are positive integers. Which they are in our case.

Therefore, I think the answer is 16. However, I recall that in some cases, the minimal sum might be higher. But according to all calculations, 16 is valid. However, I need to check if there is a possible mistake in the problem statement or my interpretation.

Wait, the problem says "the parallelogram bounded by the lines y=ax+c, y=ax+d, y=bx+c, and y=bx+d has area 18". In our case, with a = 3, b = 1, c = 3, d = 9, the lines are y = 3x + 3, y = 3x + 9, y = x + 3, y = x + 9. These are four lines forming a parallelogram. Similarly for the other case. Therefore, all conditions are satisfied.

Therefore, I think the answer is 16. However, I need to check if there is a possible mistake in my calculation of the area formula. For example, if I had a different formula. But since I verified with coordinates and shoelace formula, it's correct.

Therefore, I think the answer is 16. However, I need to check if there is a possible smaller sum. For example, if there is a case where a - b = 2, and c = 3, d = 9, but with a = 1, b = 3, which gives the same sum. Therefore, no.

Therefore, I think the answer is 16. However, I recall that in some Olympiad problems, sometimes there are hidden constraints or alternative solutions. But given the time I've spent and multiple checks, I think 16 is correct.

**Final Answer**
\boxed{16}

To solve the problem, we are given two parallelograms formed by lines with slopes $ a $ and $ b $, and constants $ c $ and $ d $. The first parallelogram is bounded by the lines:

- $ y = ax + c $
- $ y = ax + d $
- $ y = bx + c $
- $ y = bx + d $

The second parallelogram is bounded by:

- $ y = ax + c $
- $ y = ax - d $
- $ y = bx + c $
- $ y = bx - d $

We are told that the area of the first parallelogram is 18, and the area of the second is 72. We are to find the **smallest possible value of** $ a + b + c + d $, where $ a, b, c, d $ are **positive integers**.

---

### Step 1: Understanding the Area of a Parallelogram Formed by Lines

The area of a parallelogram formed by two pairs of parallel lines can be computed using the formula:

$$
\text{Area} = \frac{(c + d)^2}{|a - b|} \quad \text{or} \quad \frac{(d - c)^2}{|a - b|}
$$

depending on the configuration of the lines. In our case:

- For the **first parallelogram**, the lines are $ y = ax + c $, $ y = ax + d $, $ y = bx + c $, $ y = bx + d $, so the area is:

$$
\text{Area}_1 = \frac{(d - c)^2}{|a - b|} = 18
$$

- For the **second parallelogram**, the lines are $ y = ax + c $, $ y = ax - d $, $ y = bx + c $, $ y = bx - d $, so the area is:

$$
\text{Area}_2 = \frac{(c + d)^2}{|a - b|} = 72
$$

---

### Step 2: Relating the Two Equations

Let’s denote $ k = |a - b| $. Then we have:

$$
\frac{(d - c)^2}{k} = 18 \quad \text{and} \quad \frac{(c + d)^2}{k} = 72
$$

Dividing the second equation by the first:

$$
\frac{(c + d)^2}{(d - c)^2} = \frac{72}{18} = 4
\Rightarrow \left( \frac{c + d}{d - c} \right)^2 = 4
\Rightarrow \frac{c + d}{d - c} = 2
$$

This gives us the equation:

$$
c + d = 2(d - c)
\Rightarrow c + d = 2d - 2c
\Rightarrow 3c = d
$$

So, $ d = 3c $.

---

### Step 3: Substituting Back

Now substitute $ d = 3c $ into the first area equation:

$$
\frac{(d - c)^2}{k} = 18
\Rightarrow \frac{(3c - c)^2}{k} = 18
\Rightarrow \frac{(2c)^2}{k} = 18
\Rightarrow \frac{4c^2}{k} = 18
\Rightarrow k = \frac{2c^2}{9}
$$

Since $ k = |a - b| $, and $ a, b $ are integers, $ k $ must be a positive integer. Therefore, $ 2c^2 $ must be divisible by 9. Since 2 and 9 are coprime, $ c^2 $ must be divisible by 9, so $ c $ must be divisible by 3.

Let $ c = 3k $, then $ d = 9k $, and:

$$
k = \frac{2(3k)^2}{9} = \frac{2 \cdot 9k^2}{9} = 2k^2
$$

So, $ a - b = 2k^2 $, and we can write $ a = b + 2k^2 $.

---

### Step 4: Minimizing the Sum $ a + b + c + d $

We now express the total sum in terms of $ k $ and $ b $:

$$
a + b + c + d = (b + 2k^2) + b + 3k + 9k = 2b + 2k^2 + 12k
$$

To minimize this, we choose the smallest possible values of $ b $ and $ k $, both positive integers.

Let’s try $ k = 1 $, $ b = 1 $:

- $ c = 3k = 3 $
- $ d = 9k = 9 $
- $ a = b + 2k^2 = 1 + 2 = 3 $

So the values are:

- $ a = 3 $
- $ b = 1 $
- $ c = 3 $
- $ d = 9 $

Check the areas:

- First area: $ \frac{(d - c)^2}{a - b} = \frac{(9 - 3)^2}{3 - 1} = \frac{36}{2} = 18 $
- Second area: $ \frac{(c + d)^2}{a - b} = \frac{(3 + 9)^2}{2} = \frac{144}{2} = 72 $

Both match the given areas.

Now compute the sum:

$$
a + b + c + d = 3 + 1 + 3 + 9 = \boxed{16}
$$

---

### Final Answer

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to solve this problem about parallelograms bounded by lines and find the smallest possible value of a + b + c + d where a, b, c, d are positive integers. Let me try to break this down step by step.

First, the problem mentions two parallelograms. The first one is bounded by the lines y = ax + c, y = ax + d, y = bx + c, and y = bx + d. The area of this parallelogram is 18. The second parallelogram is bounded by y = ax + c, y = ax - d, y = bx + c, and y = bx - d, with area 72. We need to find the minimal a + b + c + d where all variables are positive integers.

Let me recall that the area of a parallelogram can be found using the base times height, but since these are parallelograms formed by lines with slopes a and b, maybe there's a formula involving the distance between the parallel lines and the angle between the lines?

Yes, I think the area of a parallelogram formed by two pairs of lines can be calculated by the product of the distances between the parallel lines and the sine of the angle between them. Let me verify that.

Suppose we have two pairs of parallel lines. The first pair is y = ax + c and y = ax + d. The distance between these two lines is |c - d| / sqrt(a² + 1). Similarly, the other pair is y = bx + c and y = bx + d, so the distance between them is |c - d| / sqrt(b² + 1). Wait, but in the first case, the lines are y = ax + c and y = ax + d, so the distance between them is |c - d| / sqrt(a² + 1). Similarly for the other pair. However, the area of the parallelogram would be the product of these two distances multiplied by the sine of the angle between the two lines?

Wait, actually, if you have two sets of parallel lines, the area of the parallelogram formed by their intersections is equal to the product of the distances between the lines divided by the sine of the angle between the lines? Wait, no, maybe it's the product of the distances multiplied by the sine of the angle between the lines?

Wait, let me think again. Let me recall that the area of a parallelogram is base times height. If I take one pair of lines as the base, then the distance between them is the height. But the other pair of lines would form the sides. Wait, maybe I need to think in terms of vectors.

Alternatively, maybe there's a formula for the area between two pairs of lines. Let me check.

Suppose we have two lines with slopes m1 and m2. The angle θ between them is given by tanθ = |(m2 - m1)/(1 + m1m2)|. Then, if we have two pairs of parallel lines, the distance between each pair is d1 and d2, then the area of the parallelogram would be d1 * d2 / sinθ? Or is it d1 * d2 * sinθ?

Wait, let me think of a simple case. Suppose we have two horizontal lines y = c and y = d, distance between them is |c - d|. Then another pair of vertical lines x = e and x = f, distance between them is |e - f|. The area of the parallelogram (which is actually a rectangle) is |c - d| * |e - f|. But in this case, the angle between the lines is 90 degrees, so sinθ = 1. So if I use the formula d1 * d2 / sinθ, that would be |c - d| * |e - f| / 1, which matches. If the angle is not 90 degrees, then maybe the formula is d1 * d2 / sinθ?

Wait, but in the case of a general parallelogram, if you have two pairs of lines with distances d1 and d2, and angle θ between them, then the area would be d1 * d2 / sinθ? Wait, no. Wait, if you have two adjacent sides of the parallelogram with lengths l and m, and angle θ between them, then the area is l * m * sinθ. But here, the distances between the lines are not the lengths of the sides. Wait, maybe I need to relate the distances between the lines to the sides of the parallelogram.

Alternatively, maybe the area can be calculated as the product of the distances between the lines divided by the sine of the angle between the lines? Wait, let me think again.

Let me take the first parallelogram. The lines y = ax + c and y = ax + d are parallel, as are y = bx + c and y = bx + d. The distance between the first pair is |c - d| / sqrt(a² + 1), and between the second pair is |c - d| / sqrt(b² + 1). The angle between the two lines is θ, where tanθ = |(b - a)/(1 + ab)|. Then, the area of the parallelogram would be the product of the distances between the lines divided by sinθ? Wait, if I think of the base as the distance between the first pair of lines, and the height as the distance between the second pair of lines divided by sinθ? Wait, maybe not.

Alternatively, if I imagine the parallelogram formed by these lines, then the sides of the parallelogram are along the directions of the lines y = ax + c and y = bx + c. The distance between the lines y = ax + c and y = ax + d is the height corresponding to the base along the direction of the lines y = bx + c. Wait, this is getting confusing. Maybe I need to use vectors.

Let me recall that the area of a parallelogram formed by two vectors is the magnitude of their cross product. So if I can find two vectors that form the sides of the parallelogram, then compute their cross product.

Alternatively, since the lines are given, maybe I can find the coordinates of the vertices of the parallelogram and then compute the area using coordinates.

Let me try that approach. Let me find the points of intersection of the lines.

First parallelogram: lines y = ax + c, y = ax + d, y = bx + c, y = bx + d.

So, the four lines are two pairs of parallel lines. The intersections will be:

Intersection of y = ax + c and y = bx + c: set ax + c = bx + c => (a - b)x = 0 => x = 0. Then y = a*0 + c = c. So point (0, c).

Similarly, intersection of y = ax + c and y = bx + d: ax + c = bx + d => (a - b)x = d - c => x = (d - c)/(a - b). Then y = a*(d - c)/(a - b) + c. Let me compute that:

y = [a(d - c) + c(a - b)] / (a - b) = [ad - ac + ac - bc]/(a - b) = [ad - bc]/(a - b). So the point is ((d - c)/(a - b), (ad - bc)/(a - b)).

Similarly, intersection of y = ax + d and y = bx + c: ax + d = bx + c => (a - b)x = c - d => x = (c - d)/(a - b) = -(d - c)/(a - b). Then y = a*(-(d - c)/(a - b)) + d = [-a(d - c) + d(a - b)]/(a - b) = [-ad + ac + ad - bd]/(a - b) = [ac - bd]/(a - b). So the point is (-(d - c)/(a - b), (ac - bd)/(a - b)).

Finally, intersection of y = ax + d and y = bx + d: ax + d = bx + d => (a - b)x = 0 => x = 0. Then y = a*0 + d = d. So point (0, d).

So the four vertices of the parallelogram are:

(0, c), ((d - c)/(a - b), (ad - bc)/(a - b)), (0, d), and (-(d - c)/(a - b), (ac - bd)/(a - b)).

Wait, but maybe I can compute the vectors between these points to find the area.

Alternatively, since it's a parallelogram, the area can be calculated by the base times height. But maybe using coordinates is more straightforward.

Alternatively, since the sides are between the lines, maybe we can compute the distance between the lines and the angle between them.

Wait, let me think again. The distance between the lines y = ax + c and y = ax + d is |c - d| / sqrt(a² + 1). Similarly, the distance between y = bx + c and y = bx + d is |c - d| / sqrt(b² + 1). The angle between the two lines is θ, where tanθ = |(b - a)/(1 + ab)|. Then, the area of the parallelogram is (distance between first pair) * (distance between second pair) / sinθ? Wait, if I think of the area as base * height, where base is the distance between the first pair of lines, and the height is the distance between the second pair of lines divided by sinθ? Wait, maybe not. Let me think.

Alternatively, if you have two vectors forming the sides of the parallelogram, then the area is the magnitude of the cross product. Let me try to find vectors.

Take two adjacent sides of the parallelogram. For example, from the point (0, c) to ((d - c)/(a - b), (ad - bc)/(a - b)). Let me compute this vector.

The vector is ((d - c)/(a - b) - 0, (ad - bc)/(a - b) - c) = ((d - c)/(a - b), (ad - bc - c(a - b))/(a - b)).

Simplify the y-component:

ad - bc - ac + bc = ad - ac = a(d - c). Therefore, the vector is ((d - c)/(a - b), a(d - c)/(a - b)) = (d - c)/(a - b) * (1, a).

Similarly, another vector from (0, c) to (-(d - c)/(a - b), (ac - bd)/(a - b)) is:

(-(d - c)/(a - b) - 0, (ac - bd)/(a - b) - c) = (-(d - c)/(a - b), (ac - bd - c(a - b))/(a - b)).

Simplify the y-component:

ac - bd - ac + bc = bc - bd = b(c - d) = -b(d - c). Therefore, the vector is (-(d - c)/(a - b), -b(d - c)/(a - b)) = -(d - c)/(a - b) * (1, b).

So the two vectors are (d - c)/(a - b)*(1, a) and -(d - c)/(a - b)*(1, b). Therefore, the area of the parallelogram is the magnitude of the cross product of these two vectors.

The cross product in 2D can be represented as the determinant:

| i    j    k |
| x1  y1   0 |
| x2  y2   0 |

Which is (x1 y2 - x2 y1) k. So the magnitude is |x1 y2 - x2 y1|.

So let me compute that.

First vector: ( (d - c)/(a - b), a(d - c)/(a - b) )

Second vector: ( -(d - c)/(a - b), -b(d - c)/(a - b) )

So x1 = (d - c)/(a - b), y1 = a(d - c)/(a - b)

x2 = -(d - c)/(a - b), y2 = -b(d - c)/(a - b)

Compute x1 y2 - x2 y1:

[(d - c)/(a - b)] * [-b(d - c)/(a - b)] - [ - (d - c)/(a - b) ] * [a(d - c)/(a - b) ]

Let me compute term by term:

First term: [(d - c)/(a - b)] * [-b(d - c)/(a - b)] = -b(d - c)^2 / (a - b)^2

Second term: - [ - (d - c)/(a - b) ] * [a(d - c)/(a - b) ] = + [ (d - c)/(a - b) * a(d - c)/(a - b) ] = a(d - c)^2 / (a - b)^2

Therefore, total is (-b + a)(d - c)^2 / (a - b)^2 = (a - b)(d - c)^2 / (a - b)^2 = (d - c)^2 / (a - b)

Therefore, the area is |(d - c)^2 / (a - b)|. Since a, b, c, d are positive integers, and the area is given as 18, so:

| (d - c)^2 / (a - b) | = 18.

But since the lines are given as y = ax + c and y = ax + d, and similarly for the other pair, we need to ensure that the denominators are not zero. Also, since a and b are slopes, they can be any real numbers, but in our case, they are positive integers. Also, since the lines are distinct, we need that c ≠ d and a ≠ b. Also, since the area is positive, the absolute value is considered. However, since the problem says that the area is 18, we can write:

(d - c)^2 / |a - b| = 18.

Similarly, for the second parallelogram, which is bounded by y = ax + c, y = ax - d, y = bx + c, and y = bx - d. Let me do the same analysis.

Following the same steps, the area would be similar. Let me check.

For the second parallelogram, the lines are y = ax + c, y = ax - d, y = bx + c, y = bx - d. So similar to the first case, but now the constants are c and -d instead of c and d. Let me compute the area.

Following the same method as before, the area would be |( (-d - c)^2 / (a - b) |? Wait, let me check.

Wait, in the first case, the difference between the constants was (d - c). In the second case, the constants are c and -d. So the difference between the constants for the first pair of lines (y = ax + c and y = ax - d) is c - (-d) = c + d. Similarly, for the second pair of lines (y = bx + c and y = bx - d), the difference is c - (-d) = c + d. Therefore, following the same formula as before, the area would be (c + d)^2 / |a - b| = 72.

Wait, let me verify this. Let me go through the same steps as before for the second parallelogram.

Let me find the vectors again. The lines are y = ax + c, y = ax - d, y = bx + c, y = bx - d.

Following similar steps as before, the points of intersection would be:

Intersection of y = ax + c and y = bx + c: (0, c) as before.

Intersection of y = ax + c and y = bx - d: ax + c = bx - d => (a - b)x = -d - c => x = -(c + d)/(a - b). Then y = a*(-(c + d)/(a - b)) + c = [-a(c + d) + c(a - b)]/(a - b) = [-ac - ad + ac - bc]/(a - b) = (-ad - bc)/(a - b) = -(ad + bc)/(a - b). So the point is (-(c + d)/(a - b), -(ad + bc)/(a - b)).

Similarly, intersection of y = ax - d and y = bx + c: ax - d = bx + c => (a - b)x = c + d => x = (c + d)/(a - b). Then y = a*(c + d)/(a - b) - d = [a(c + d) - d(a - b)]/(a - b) = [ac + ad - ad + bd]/(a - b) = (ac + bd)/(a - b). So the point is ((c + d)/(a - b), (ac + bd)/(a - b)).

Intersection of y = ax - d and y = bx - d: ax - d = bx - d => (a - b)x = 0 => x = 0. Then y = a*0 - d = -d. So point (0, -d).

So the four vertices are (0, c), (-(c + d)/(a - b), -(ad + bc)/(a - b)), (0, -d), and ((c + d)/(a - b), (ac + bd)/(a - b)).

Similarly, computing vectors from (0, c) to ((c + d)/(a - b), (ac + bd)/(a - b)):

x-component: (c + d)/(a - b) - 0 = (c + d)/(a - b)

y-component: (ac + bd)/(a - b) - c = [ac + bd - c(a - b)]/(a - b) = [ac + bd - ac + bc]/(a - b) = (bd + bc)/(a - b) = b(d + c)/(a - b)

So the vector is (c + d)/(a - b) * (1, b)

Similarly, the vector from (0, c) to (-(c + d)/(a - b), -(ad + bc)/(a - b)) is:

x-component: -(c + d)/(a - b) - 0 = -(c + d)/(a - b)

y-component: -(ad + bc)/(a - b) - c = [ -ad - bc - c(a - b) ] / (a - b) = [ -ad - bc - ac + bc ] / (a - b) = (-ad - ac)/ (a - b) = -a(d + c)/(a - b)

So the vector is -(c + d)/(a - b) * (1, a)

Therefore, the cross product of these two vectors is:

First vector: ( (c + d)/(a - b), b(c + d)/(a - b) )

Second vector: ( -(c + d)/(a - b), -a(c + d)/(a - b) )

So cross product determinant:

x1 y2 - x2 y1 = [ (c + d)/(a - b) * (-a(c + d)/(a - b)) ] - [ -(c + d)/(a - b) * b(c + d)/(a - b) ]

= [ -a(c + d)^2 / (a - b)^2 ] - [ -b(c + d)^2 / (a - b)^2 ]

= [ -a + b ] (c + d)^2 / (a - b)^2

= (b - a)(c + d)^2 / (a - b)^2

= -(a - b)(c + d)^2 / (a - b)^2

= - (c + d)^2 / (a - b)

Taking absolute value, the area is | - (c + d)^2 / (a - b) | = (c + d)^2 / |a - b|.

Therefore, the area of the second parallelogram is (c + d)^2 / |a - b| = 72.

So now, we have two equations:

From the first parallelogram: (d - c)^2 / |a - b| = 18.

From the second parallelogram: (c + d)^2 / |a - b| = 72.

Let me denote |a - b| as k. Since a and b are positive integers, and assuming a ≠ b (since otherwise the lines would be parallel and not form a parallelogram). Also, since the denominators are positive, we can write:

Let me assume that a > b, so that a - b is positive. Then |a - b| = a - b. So:

First equation: (d - c)^2 / (a - b) = 18.

Second equation: (c + d)^2 / (a - b) = 72.

Therefore, if I divide the second equation by the first equation, I get:

[(c + d)^2 / (a - b)] / [(d - c)^2 / (a - b)] = 72 / 18 => [(c + d)^2] / [(d - c)^2] = 4 => [(c + d)/(d - c)]^2 = 4 => (c + d)/(d - c) = ±2.

But since c and d are positive integers, and d - c must be positive (since otherwise the denominator would be negative or zero, but since we have (d - c)^2 in the first equation, which is positive, but if d - c is negative, then (d - c)^2 is same as (c - d)^2. However, in our case, since we have (d - c)^2 / (a - b) = 18, and a - b is positive, so (d - c)^2 must be positive, so d ≠ c. Also, since the lines are distinct, so d ≠ c. However, the sign of (d - c) depends on whether d > c or c > d.

But in the ratio [(c + d)/(d - c)]^2 = 4, taking square roots, we have (c + d)/|d - c| = 2. Because the square is 4, so the absolute value is 2. Therefore:

(c + d)/|d - c| = 2.

Therefore, c + d = 2 |d - c|.

Let me consider two cases:

Case 1: d > c. Then |d - c| = d - c. Therefore:

c + d = 2(d - c) => c + d = 2d - 2c => c + d - 2d + 2c = 0 => 3c - d = 0 => d = 3c.

Case 2: c > d. Then |d - c| = c - d. Therefore:

c + d = 2(c - d) => c + d = 2c - 2d => c + d - 2c + 2d = 0 => -c + 3d = 0 => c = 3d.

So in either case, we have that one of the variables is three times the other. So either d = 3c or c = 3d.

But since c and d are positive integers, both possibilities are valid. Let me note that.

So, let me consider both cases.

Case 1: d = 3c.

Then, substituting into the first equation:

(d - c)^2 / (a - b) = 18 => (3c - c)^2 / (a - b) = 18 => (2c)^2 / (a - b) = 18 => 4c² / (a - b) = 18 => 2c² / (a - b) = 9 => a - b = 2c² / 9.

Similarly, from the second equation:

(c + d)^2 / (a - b) = 72 => (c + 3c)^2 / (a - b) = 72 => (4c)^2 / (a - b) = 72 => 16c² / (a - b) = 72 => 2c² / (a - b) = 9, which is the same as the first equation. So that's consistent.

Therefore, in this case, a - b = 2c² / 9. Since a and b are positive integers, a - b must be a positive integer (since we assumed a > b). Therefore, 2c² must be divisible by 9. Therefore, 9 divides 2c². Since 2 and 9 are coprime, 9 divides c², so 3 divides c. Let me write c = 3k, where k is a positive integer. Then:

c = 3k, so d = 3c = 9k.

Then, a - b = 2*(3k)^2 / 9 = 2*9k² / 9 = 2k².

Therefore, a - b = 2k². Since a and b are positive integers, and a > b, we can write a = b + 2k².

Now, our variables are:

c = 3k,

d = 9k,

a = b + 2k².

We need to find positive integers a, b, c, d with these relations, and minimize a + b + c + d.

Similarly, let's check Case 2: c = 3d.

Then, substituting into the first equation:

(d - c)^2 / (a - b) = 18 => (d - 3d)^2 / (a - b) = 18 => (-2d)^2 / (a - b) = 18 => 4d² / (a - b) = 18 => 2d² / (a - b) = 9 => a - b = 2d² / 9.

Similarly, second equation:

(c + d)^2 / (a - b) = 72 => (3d + d)^2 / (a - b) = 72 => (4d)^2 / (a - b) = 72 => 16d² / (a - b) = 72 => 2d² / (a - b) = 9, same as before.

Therefore, similar to Case 1, a - b = 2d² / 9. Therefore, 2d² must be divisible by 9, so 9 divides 2d², so 3 divides d. Let me write d = 3m, where m is a positive integer. Then c = 3d = 9m.

Then, a - b = 2*(3m)^2 / 9 = 2*9m² / 9 = 2m². Therefore, a = b + 2m².

So variables:

d = 3m,

c = 9m,

a = b + 2m².

Again, need to minimize a + b + c + d = (b + 2m²) + b + 9m + 3m = 2b + 2m² + 12m.

Similarly, in Case 1, variables:

c = 3k,

d = 9k,

a = b + 2k²,

so a + b + c + d = (b + 2k²) + b + 3k + 9k = 2b + 2k² + 12k.

So both cases have similar expressions. Therefore, the minimal value of a + b + c + d would be the same in both cases, depending on k or m. However, since c and d are positive integers, and k and m are positive integers, we can consider both cases and find the minimal value.

But since the problem says that a, b, c, d are positive integers, and we need to find the minimal a + b + c + d, we can consider both cases and find the minimal possible value.

But perhaps there's a way to combine both cases. However, since in Case 1, c = 3k and d = 9k, and in Case 2, c = 9m and d = 3m, so they are symmetric. Therefore, the minimal value would be the same in both cases, so we can just consider one case and find the minimal value.

So let me focus on Case 1: c = 3k, d = 9k, a = b + 2k². Then, the total sum is 2b + 2k² + 12k. To minimize this, we need to choose k and b as small as possible.

But note that a and b must be positive integers. Since a = b + 2k², and b must be a positive integer, so b ≥ 1. Therefore, the minimal value of b is 1, and then a = 1 + 2k². Therefore, the total sum is 2*1 + 2k² + 12k = 2 + 2k² + 12k.

Similarly, for Case 2, the total sum is 2b + 2m² + 12m, with similar reasoning. Therefore, regardless of the case, the minimal sum would be achieved when k or m is as small as possible, and b is as small as possible.

But we need to check if there are any constraints on a and b. For example, in Case 1, a = b + 2k². Since a and b are positive integers, and b ≥ 1, then a is automatically positive. However, we need to ensure that the lines are distinct and that the denominators are non-zero. But since a ≠ b (as a - b = 2k² > 0), and c and d are positive integers, so all conditions are satisfied.

Therefore, to minimize the total sum, we need to minimize 2b + 2k² + 12k. Since b ≥ 1, the minimal value occurs when b = 1, and k is as small as possible. Let's check for k = 1:

Case 1, k = 1:

c = 3*1 = 3,

d = 9*1 = 9,

a = b + 2*(1)^2 = b + 2.

Total sum: 2b + 2*(1)^2 + 12*1 = 2b + 2 + 12 = 2b + 14. Since b ≥ 1, minimal when b = 1: 2*1 + 14 = 16. So total sum is 16 + a + ... Wait, no, wait:

Wait, the total sum is a + b + c + d = (b + 2k²) + b + c + d = 2b + 2k² + c + d. Wait, earlier I said 2b + 2k² + 12k. Wait, c = 3k, d = 9k, so c + d = 12k. Therefore, total sum is 2b + 2k² + 12k. So for k = 1, b = 1: 2*1 + 2*1 + 12*1 = 2 + 2 + 12 = 16. So total sum is 16. But let me check if this is valid.

Wait, a = b + 2k² = 1 + 2*1 = 3. So a = 3, b = 1, c = 3, d = 9. Let me check if this satisfies the original area conditions.

First parallelogram area: (d - c)^2 / (a - b) = (9 - 3)^2 / (3 - 1) = 36 / 2 = 18. Correct.

Second parallelogram area: (c + d)^2 / (a - b) = (3 + 9)^2 / 2 = 144 / 2 = 72. Correct. So this works. Therefore, the total sum is a + b + c + d = 3 + 1 + 3 + 9 = 16. Wait, but 3 + 1 + 3 + 9 is 16? 3 + 1 is 4, 3 + 9 is 12, total 16. Yes. So that's a possible value. But the problem says that a, b, c, d are positive integers. Here, a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid. But is this the minimal possible?

Wait, but maybe there are other cases where k is larger but b is smaller? Wait, no, since b is at least 1. However, maybe there are other cases where the ratio is different? Wait, but we considered both cases where d = 3c and c = 3d, and found that in both cases, the minimal sum is 16. However, let me check if there are other possibilities.

Wait, but maybe there are other solutions where the ratio (c + d)/|d - c| is 2, but with different relations? Wait, no, we considered both cases where d > c and c > d, leading to d = 3c or c = 3d. So those are the only possibilities.

But let me check if there are other possibilities where the ratio is -2? Wait, no, because we took absolute value. So the ratio is 2. Therefore, only those two cases.

But let me check if there are other solutions where the equations hold. For example, suppose that (c + d)^2 / (a - b) = 72 and (d - c)^2 / (a - b) = 18. Let me denote (d - c)^2 = 18 * (a - b) and (c + d)^2 = 72 * (a - b). Therefore, dividing the second equation by the first, we get [(c + d)^2]/[(d - c)^2] = 4, which leads to the same ratio as before. Therefore, the only possibilities are those two cases. Therefore, the minimal sum is 16? But let me check if there are smaller values.

Wait, let me check for k = 1, b = 1 gives total sum 16. Is there a way to get a smaller sum?

Suppose k = 1, b = 1: sum is 16.

What if k = 1, b = 1, but maybe there's another case where k is smaller? But k is a positive integer, so k cannot be smaller than 1.

Alternatively, maybe there are other solutions where the equations hold with different relations? For example, if (c + d)^2 / (a - b) = 72 and (d - c)^2 / (a - b) = 18, but with different relations between c and d?

Wait, but we already derived that the only possibilities are d = 3c or c = 3d. Therefore, there are no other possibilities. Therefore, the minimal sum is 16? But let me check if there are other solutions with different k and b.

For example, take k = 1, b = 1: sum 16.

If k = 1, b = 2: sum is 2*2 + 2*1 + 12*1 = 4 + 2 + 12 = 18, which is larger.

Similarly, for k = 2, b = 1: sum is 2*1 + 2*(2)^2 + 12*2 = 2 + 8 + 24 = 34, which is larger.

Alternatively, if we take Case 2, where c = 9m and d = 3m. Let me check for m = 1:

c = 9*1 = 9, d = 3*1 = 3. Then, a - b = 2*(3m)^2 / 9 = 2*(9)/9 = 2. Therefore, a = b + 2. Then total sum is a + b + c + d = (b + 2) + b + 9 + 3 = 2b + 14. Again, minimal when b = 1: 2 + 14 = 16. Which is same as before. So same result.

Therefore, regardless of case, the minimal sum is 16. However, let me check if there are other possibilities where the equations hold with different relations. For example, if the ratio [(c + d)/(d - c)] is 2, but with different values?

Wait, but we already considered that. So unless there are other solutions where [(c + d)/(d - c)] is 2, but with different relations. But no, because we derived that only d = 3c or c = 3d.

But let me check if there are other solutions where the equations hold. For example, suppose that (d - c)^2 / (a - b) = 18 and (c + d)^2 / (a - b) = 72. Let me denote x = d - c and y = c + d. Then, we have:

x² / k = 18,

y² / k = 72,

where k = a - b.

Therefore, y² / x² = 72 / 18 = 4 => y / x = 2 => y = 2x.

But y = c + d and x = d - c. Therefore:

c + d = 2(d - c) => c + d = 2d - 2c => 3c = d, which is Case 1.

Or, if we consider y = -2x, but since y and x are positive? Wait, x = d - c. If d < c, then x is negative, but since we have x², it's same as (c - d)^2. However, in the ratio y / x, if x is negative, then y / x = -2. But since y = c + d is positive, and x = d - c could be negative. However, since we have y² / x² = 4, so y / |x| = 2. Therefore, y = 2|x|. Therefore, if x = d - c is negative, then |x| = c - d, so y = 2(c - d). Therefore:

c + d = 2(c - d) => c + d = 2c - 2d => -c + 3d = 0 => c = 3d, which is Case 2.

Therefore, only those two cases. Therefore, there are no other solutions. Therefore, the minimal sum is 16. However, let me check if there are other solutions where k is not integer? Wait, no, because in our equations, we have that a - b = 2k², and since a and b are integers, then 2k² must be integer, which it is, since k is integer. Therefore, all solutions are covered in these cases.

But wait, the problem says that a, b, c, d are positive integers. In our case, with k = 1, we have a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid. However, let me check if there are smaller values. For example, if we take k = 1, but maybe different b?

Wait, if we take b = 1, then a = 3. If we take b = 0, but b must be positive integer, so b cannot be 0. Therefore, the minimal b is 1. Therefore, 16 is the minimal sum? But let me check if there are other possibilities where the equations hold with different relations. For example, if the ratio is different?

Wait, but we derived that [(c + d)/(d - c)]^2 = 4, so the ratio is 2. Therefore, no other possibilities. Therefore, 16 is the minimal sum. However, let me check if there are other solutions with different k and b where the total sum is less than 16.

For example, suppose k = 1, b = 1 gives sum 16. What if there is another solution where k is not integer? But no, since k is defined as c = 3k, so k must be integer. Therefore, no. Therefore, 16 is the minimal sum. However, let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of c and d?

Wait, for example, suppose that (d - c)^2 = 18 * k and (c + d)^2 = 72 * k. Therefore, (c + d)^2 = 4*(d - c)^2. Therefore, c + d = 2(d - c) or c + d = -2(d - c). Which leads to the same cases as before. Therefore, no other solutions.

But let me check with numbers. Suppose that c = 1, d = 3. Then, d - c = 2, c + d = 4. Then, 4^2 / (a - b) = 72 => 16 / (a - b) = 72 => a - b = 16 / 72 = 2/9. Not integer. Therefore, invalid. Similarly, if c = 2, d = 6. Then d - c = 4, c + d = 8. Then 8^2 / (a - b) = 72 => 64 / (a - b) = 72 => a - b = 64 / 72 = 8/9. Not integer. Similarly, c = 3, d = 9: d - c = 6, c + d = 12. Then 12² / (a - b) = 72 => 144 / (a - b) = 72 => a - b = 2. Which is valid. Which is our previous case. Therefore, this gives a - b = 2. Which is the case when k = 1, since c = 3k = 3*1 = 3, d = 9k = 9*1 = 9. Therefore, this is the same as before.

Similarly, if c = 1, d = 3, but then a - b = 2c² / 9 = 2*1 / 9 = 2/9, which is not integer. Therefore, invalid. Therefore, only when c is multiple of 3, then 2c² / 9 is integer. Therefore, c must be multiple of 3. Therefore, the minimal c is 3, leading to k = 1. Therefore, the minimal sum is 16.

But wait, the problem says that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid. However, let me check if there is another combination with smaller sum. For example, if we take k = 1, but different b? For example, if b = 1, sum is 16. If b = 2, sum is 2*2 + 2*1 + 12*1 = 4 + 2 + 12 = 18. Which is larger. If we take k = 1, b = 1, sum is 16. If we take k = 1, b = 1, and check if there are other variables. For example, a = 3, b = 1, c = 3, d = 9. All positive integers. So that's valid.

But let me check if there are other solutions where the equations hold with different relations. For example, suppose that (d - c)^2 / (a - b) = 18 and (c + d)^2 / (a - b) = 72, but with different values of c and d. For example, suppose that (d - c) = 3 and (c + d) = 6. Then, 6² / (a - b) = 72 => 36 / (a - b) = 72 => a - b = 0.5. Not integer. Not valid. If (d - c) = 6, (c + d) = 12, then 12² / (a - b) = 72 => 144 / (a - b) = 72 => a - b = 2. Which is our previous case. If (d - c) = 9, (c + d) = 18, then 18² / (a - b) = 72 => 324 / (a - b) = 72 => a - b = 324 / 72 = 4.5. Not integer. Not valid. If (d - c) = 12, (c + d) = 24, then 24² / (a - b) = 72 => 576 / (a - b) = 72 => a - b = 8. Then, (d - c) = 12, so d = c + 12. Also, c + d = 24 => c + c + 12 = 24 => 2c = 12 => c = 6, d = 18. Then, a - b = 8. Then, total sum is a + b + c + d = (b + 8) + b + 6 + 18 = 2b + 32. Minimal when b = 1: 2 + 32 = 34, which is larger than 16. Therefore, not better.

Alternatively, if (d - c) = 2, then (c + d) = 4. Then, 4² / (a - b) = 72 => 16 / (a - b) = 72 => a - b = 16 / 72 = 2/9. Not integer. Not valid.

Therefore, the only valid solutions are when (d - c) is multiple of 6, leading to c = 3k, d = 9k. Therefore, the minimal sum is 16. However, let me check if there are other possibilities where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of (d - c) and (c + d). For example, if (d - c) = 6, then (c + d) = 12, which gives us the case we already considered. If (d - c) = 3, then (c + d) = 6, but that leads to a - b = 2/9, which is invalid. Similarly, (d - c) = 9, (c + d) = 18, which gives a - b = 4.5, invalid. Therefore, only multiples where (d - c) is multiple of 6, leading to c = 3k, d = 9k.

Therefore, the minimal sum is 16. However, let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of a and b. For example, suppose that a - b is different. Wait, but we have that a - b is determined by the equations. For example, in the case where c = 3k, d = 9k, then a - b = 2k². Therefore, if k = 1, a - b = 2. If k = 2, a - b = 8, etc. Therefore, the minimal a - b is 2, which gives us the minimal sum.

But let me check if there are other solutions where a - b is different. For example, suppose that (d - c)^2 = 18 * (a - b) and (c + d)^2 = 72 * (a - b). Let me suppose that (d - c) = 6 and (c + d) = 12, which gives a - b = 2. Which is our case. Alternatively, if (d - c) = 12 and (c + d) = 24, then a - b = 144 / 72 = 2. Wait, no:

Wait, (c + d)^2 / (a - b) = 72. If (c + d) = 24, then 24² / (a - b) = 72 => 576 / (a - b) = 72 => a - b = 576 / 72 = 8. But (d - c) = 12, so (d - c)^2 / (a - b) = 144 / 8 = 18. Which is correct. Therefore, this is another solution. In this case, c + d = 24, d - c = 12. Therefore, solving:

c + d = 24,

d - c = 12.

Adding equations: 2d = 36 => d = 18, then c = 24 - 18 = 6. Therefore, c = 6, d = 18. Then, a - b = 8. Then, total sum is a + b + c + d = (b + 8) + b + 6 + 18 = 2b + 32. Minimal when b = 1: 2 + 32 = 34. Which is larger than 16. Therefore, not better.

Therefore, the minimal sum is indeed 16. However, let me check if there are other solutions with smaller sum. For example, if we take k = 1, but different values of a and b? Wait, no, because a = b + 2k². If k = 1, then a = b + 2. Therefore, if b = 1, a = 3. If b = 2, a = 4, etc. Therefore, the minimal sum occurs when b is minimal, which is 1. Therefore, 16 is the minimal sum.

But let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of c and d. For example, suppose that (d - c) = 6 and (c + d) = 12, which gives us c = 3, d = 9. Which is our case. Alternatively, if (d - c) = 6 and (c + d) = 12, but with different c and d? No, because solving c + d = 12 and d - c = 6 gives unique solution c = 3, d = 9. Therefore, no other solutions.

Alternatively, if (d - c) = 6 and (c + d) = 12, then same as above. Therefore, no other solutions. Therefore, the minimal sum is 16.

But wait, the problem says that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9. All positive integers. However, let me check if there are other solutions with smaller sum. For example, if we take k = 1, but with different b? No, since b is minimal 1. Therefore, 16 is the minimal sum.

But let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of a and b. For example, suppose that a - b = 1. Then, from the first equation: (d - c)^2 = 18 * 1 = 18. Therefore, d - c = sqrt(18) = 3√2, which is not integer. Therefore, invalid. Similarly, a - b = 2: (d - c)^2 = 36 => d - c = 6. Then, (c + d)^2 = 72 * 2 = 144 => c + d = 12. Which gives c = 3, d = 9. Which is our case. If a - b = 3: (d - c)^2 = 18 * 3 = 54. Not a perfect square. Similarly, a - b = 4: (d - c)^2 = 72. Not a perfect square. a - b = 6: (d - c)^2 = 108. Not a perfect square. a - b = 8: (d - c)^2 = 144 => d - c = 12. Then, (c + d)^2 = 72 * 8 = 576 => c + d = 24. Which gives c = 6, d = 18. Which is another solution, but sum is higher. Therefore, the minimal a - b is 2, leading to the minimal sum of 16.

Therefore, the answer should be 16. However, let me check if there are other solutions with a - b = 2 and different c and d. For example, if a - b = 2, then (d - c)^2 = 18 * 2 = 36 => d - c = 6. And (c + d)^2 = 72 * 2 = 144 => c + d = 12. Therefore, solving:

c + d = 12,

d - c = 6.

Adding equations: 2d = 18 => d = 9, then c = 3. Therefore, unique solution. Therefore, only one solution for a - b = 2. Therefore, the minimal sum is 16.

But let me check if there are other solutions where a - b is different but the total sum is smaller. For example, if a - b = 2, sum is 16. If a - b = something else, but with smaller total sum? For example, if a - b = 2, and c = 3, d = 9, sum is 16. If there is another solution with a - b = 2, but different c and d? No, because c and d are uniquely determined by the equations. Therefore, no.

Therefore, the minimal possible value of a + b + c + d is 16. However, let me check the problem statement again to make sure I didn't miss anything.

Wait, the problem says that the parallelogram bounded by y = ax + c, y = ax + d, y = bx + c, y = bx + d has area 18. The other one has area 72. Given that a, b, c, d are positive integers. What is the smallest possible value of a + b + c + d?

Yes, and in our case, a = 3, b = 1, c = 3, d = 9. All positive integers. Sum is 3 + 1 + 3 + 9 = 16. However, let me check if there are other solutions with smaller sum. For example, if there is a case where c and d are smaller. For example, if c = 1, d = 3, but then (d - c)^2 / (a - b) = 4 / (a - b) = 18 => a - b = 4 / 18 = 2/9, which is not integer. Similarly, c = 2, d = 6: (6 - 2)^2 / (a - b) = 16 / (a - b) = 18 => a - b = 16 / 18 = 8/9, not integer. c = 3, d = 9: gives a - b = 2, which is valid. Therefore, no smaller c and d possible. Therefore, 16 is the minimal sum.

But wait, the problem says that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9. All positive integers. However, let me check if there is another solution where a and b are swapped? For example, if a = 1, b = 3. But then a - b = -2, but we took absolute value earlier. However, in our equations, we assumed a > b, but if a < b, then |a - b| = b - a. However, in our equations, we had:

From the first parallelogram: (d - c)^2 / |a - b| = 18.

From the second: (c + d)^2 / |a - b| = 72.

Therefore, even if a < b, the equations are same. Therefore, if we take a = 1, b = 3, then |a - b| = 2, same as before. Then, same equations. Therefore, c = 3, d = 9. Then, a + b + c + d = 1 + 3 + 3 + 9 = 16. Same sum. Therefore, same result.

But in this case, a = 1, b = 3, c = 3, d = 9. All positive integers. Therefore, same sum. Therefore, regardless of which is larger between a and b, the sum remains same.

But let me check if there are other solutions with different a and b. For example, if a - b = 2, then a = b + 2. If we take b = 2, then a = 4. Then, c = 3, d = 9. Then, sum is 4 + 2 + 3 + 9 = 18, which is larger than 16. Therefore, not better.

Therefore, the minimal sum is indeed 16. However, let me check if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of c and d. For example, if c = 6, d = 18, then (d - c)^2 / (a - b) = 144 / (a - b) = 18 => a - b = 8. Then, (c + d)^2 / (a - b) = 24² / 8 = 576 / 8 = 72. Correct. Therefore, this is another solution. Then, a - b = 8. If we take b = 1, then a = 9. Then, sum is 9 + 1 + 6 + 18 = 34. Which is larger than 16. Therefore, not better.

Therefore, the minimal sum is 16. However, let me check if there are other solutions with smaller sum. For example, if we take k = 1, but different values of a and b? No, because a = b + 2. Therefore, minimal when b = 1. Therefore, 16 is the minimal.

But wait, the problem says that a, b, c, d are positive integers. In our case, c = 3 and d = 9. However, what if there are other solutions where c and d are smaller? For example, if c = 3, d = 9, but with different a and b? No, because a - b is fixed once c and d are fixed. Therefore, no.

Alternatively, if there are other solutions where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of (d - c) and (c + d). For example, if (d - c) = 6 and (c + d) = 12, which gives us the case we already considered. Therefore, no other solutions.

Therefore, I think the minimal sum is 16. However, I need to check if there are any constraints that I missed. For example, the problem says that the lines are y = ax + c, y = ax + d, etc. Are there any restrictions on the slopes? For example, if a and b are positive integers, and the lines are distinct. Since a and b are positive integers, and we have a ≠ b (since a - b is non-zero). Therefore, all conditions are satisfied.

Therefore, I think the answer is 16. However, let me check once again with the given values:

a = 3, b = 1, c = 3, d = 9.

First parallelogram: lines y = 3x + 3, y = 3x + 9, y = 1x + 3, y = 1x + 9.

Area is 18. Let me compute it using the formula we derived: (d - c)^2 / |a - b| = (9 - 3)^2 / |3 - 1| = 36 / 2 = 18. Correct.

Second parallelogram: lines y = 3x + 3, y = 3x - 9, y = 1x + 3, y = 1x - 9.

Area is (c + d)^2 / |a - b| = (3 + 9)^2 / 2 = 144 / 2 = 72. Correct.

Therefore, all conditions are satisfied. Therefore, the minimal sum is 16. However, the problem says "the smallest possible value of a + b + c + d". But let me check if there is a case where a, b, c, d are smaller. For example, if there is a case where a = 1, b = 3, c = 3, d = 9, which gives the same sum. Or if there is another combination with smaller numbers.

Wait, let me think. Suppose that instead of c = 3, d = 9, we have c = 1, d = 3, but then as before, a - b = 2/9, which is invalid. Therefore, no. Therefore, 16 is the minimal.

But wait, the problem says that a, b, c, d are positive integers. In our case, c = 3 and d = 9. However, what if there is a case where c and d are smaller? For example, if c = 3, d = 9, but with different a and b? No, because a - b is fixed. Therefore, no.

Alternatively, if there is a case where the equations hold with different relations. For example, if the first area is 18 and the second is 72, but with different values of (d - c) and (c + d). For example, if (d - c) = 6 and (c + d) = 12, which gives us the case we already considered. Therefore, no.

Therefore, I think the answer is 16. However, I need to check if there is a possible mistake in my calculation. For example, when I calculated the area of the parallelogram, I used the formula (d - c)^2 / |a - b|. Let me verify this with coordinates.

Take the first parallelogram with a = 3, b = 1, c = 3, d = 9. The four vertices are:

(0, 3), ((d - c)/(a - b), (ad - bc)/(a - b)) = (6 / 2, (27 - 3)/2) = (3, 12), (0, 9), and (-3, (ac - bd)/(a - b)) = (-3, (9 - 27)/2) = (-3, -9). Wait, let me compute the coordinates again.

Wait, earlier, we had for the first parallelogram:

Vertices are (0, c) = (0, 3), ((d - c)/(a - b), (ad - bc)/(a - b)) = ( (9 - 3)/(3 - 1), (3*9 - 1*3)/(3 - 1) ) = (6/2, (27 - 3)/2) = (3, 24/2) = (3, 12). Then (0, d) = (0, 9), and the fourth vertex is (-(d - c)/(a - b), (ac - bd)/(a - b)) = (-3, (3*3 - 1*9)/2) = (-3, (9 - 9)/2) = (-3, 0). Wait, that's different from what I wrote earlier. Wait, earlier I thought the fourth vertex was (-3, -9), but that seems incorrect. Let me recalculate.

Wait, the fourth vertex is intersection of y = ax + d and y = bx + c. Which we calculated earlier as ((c - d)/(a - b), (ac - bd)/(a - b)). Wait, no:

Wait, earlier, for the first parallelogram, the intersection of y = ax + d and y = bx + c is:

ax + d = bx + c => (a - b)x = c - d => x = (c - d)/(a - b) = -(d - c)/(a - b). Then y = a*x + d = a*(-(d - c)/(a - b)) + d = [-a(d - c) + d(a - b)] / (a - b) = [-ad + ac + ad - bd]/(a - b) = [ac - bd]/(a - b). Therefore, the point is (-(d - c)/(a - b), (ac - bd)/(a - b)).

In our case, a = 3, b = 1, c = 3, d = 9. Therefore:

x = -(9 - 3)/(3 - 1) = -6/2 = -3.

y = (3*3 - 1*9)/(3 - 1) = (9 - 9)/2 = 0/2 = 0. Therefore, the fourth vertex is (-3, 0). Therefore, the four vertices are (0, 3), (3, 12), (0, 9), (-3, 0). Wait, but how does this form a parallelogram?

Let me plot these points:

(0, 3), (3, 12), (0, 9), (-3, 0). Wait, connecting these points:

From (0, 3) to (3, 12): this is one side.

From (0, 3) to (-3, 0): another side.

From (3, 12) to (0, 9): another side.

From (0, 9) to (-3, 0): another side.

Wait, but to compute the area, maybe using coordinates. Let me use the shoelace formula.

List the coordinates in order:

Let me take the points in order: (0, 3), (3, 12), (0, 9), (-3, 0), back to (0, 3).

Shoelace formula:

Area = 1/2 |sum_{i=1 to n} (x_i y_{i+1} - x_{i+1} y_i)|

Compute each term:

First term: x1 y2 - x2 y1 = 0*12 - 3*3 = 0 - 9 = -9

Second term: x2 y3 - x3 y2 = 3*9 - 0*12 = 27 - 0 = 27

Third term: x3 y4 - x4 y3 = 0*0 - (-3)*9 = 0 + 27 = 27

Fourth term: x4 y1 - x1 y4 = (-3)*3 - 0*0 = -9 - 0 = -9

Sum: -9 + 27 + 27 - 9 = 36

Area = 1/2 * |36| = 18. Correct. Therefore, the area is indeed 18. Similarly, for the second parallelogram, let me check.

Second parallelogram: lines y = 3x + 3, y = 3x - 9, y = 1x + 3, y = 1x - 9.

Vertices:

Intersection of y = 3x + 3 and y = x + 3: (0, 3).

Intersection of y = 3x + 3 and y = x - 9: 3x + 3 = x - 9 => 2x = -12 => x = -6. y = 3*(-6) + 3 = -18 + 3 = -15. So point (-6, -15).

Intersection of y = 3x - 9 and y = x + 3: 3x - 9 = x + 3 => 2x = 12 => x = 6. y = 6 + 3 = 9. So point (6, 9).

Intersection of y = 3x - 9 and y = x - 9: 3x - 9 = x - 9 => 2x = 0 => x = 0. y = -9. So point (0, -9).

Therefore, the four vertices are (0, 3), (-6, -15), (0, -9), (6, 9). Let me apply shoelace formula.

Order of points: (0, 3), (-6, -15), (0, -9), (6, 9), back to (0, 3).

Compute terms:

First term: x1 y2 - x2 y1 = 0*(-15) - (-6)*3 = 0 + 18 = 18

Second term: x2 y3 - x3 y2 = (-6)*(-9) - 0*(-15) = 54 - 0 = 54

Third term: x3 y4 - x4 y3 = 0*9 - 6*(-9) = 0 + 54 = 54

Fourth term: x4 y1 - x1 y4 = 6*3 - 0*9 = 18 - 0 = 18

Sum: 18 + 54 + 54 + 18 = 144

Area = 1/2 * |144| = 72. Correct. Therefore, the area is indeed 72. Therefore, the values are correct.

Therefore, the minimal sum is 16. However, I need to check if there is a case where the sum is smaller. For example, if there is a case where a, b, c, d are smaller. For example, if a = 2, b = 1, c = ?, d = ?. Let me check.

Suppose a = 2, b = 1. Then, |a - b| = 1. Then, from the first equation: (d - c)^2 / 1 = 18 => (d - c)^2 = 18. Not possible since 18 is not a perfect square. Therefore, invalid.

If a = 4, b = 2, then |a - b| = 2. Then, (d - c)^2 = 36 => d - c = 6. And (c + d)^2 = 72 * 2 = 144 => c + d = 12. Therefore, solving c + d = 12, d - c = 6 => d = 9, c = 3. Therefore, a = 4, b = 2, c = 3, d = 9. Sum is 4 + 2 + 3 + 9 = 18, which is larger than 16.

If a = 2, b = 1, no solution. If a = 1, b = 3, then |a - b| = 2. Then, same as before, c = 3, d = 9. Sum is 1 + 3 + 3 + 9 = 16. Same as before.

Therefore, no smaller sum. Therefore, the answer is 16.

But wait, the problem says "the smallest possible value of a + b + c + d". However, I need to check if there are other solutions with different a, b, c, d. For example, if a = 1, b = 3, c = 3, d = 9, sum is 16. If a = 1, b = 3, c = 3, d = 9, or a = 3, b = 1, c = 3, d = 9. Both give same sum. Are there other combinations?

For example, if a = 1, b = 3, c = 3, d = 9. Or if a = 1, b = 3, c = 3, d = 9. Or if a = 1, b = 3, c = 3, d = 9. No other combinations. Therefore, 16 is the minimal.

But let me check if there are solutions where c and d are swapped. For example, if c = 9, d = 3. Then, in the first case, (d - c)^2 = (3 - 9)^2 = 36, same as before. Then, (c + d)^2 = (9 + 3)^2 = 144. Therefore, same equations. Therefore, a - b = 2. Therefore, a = b + 2. Then, if b = 1, a = 3. Therefore, sum is 3 + 1 + 9 + 3 = 16. Same as before. Therefore, same result.

Therefore, regardless of the order of c and d, the sum remains same. Therefore, 16 is the minimal sum.

However, I need to check if there are other solutions where the equations hold with different values. For example, if a - b = 2, and c = 3, d = 9, or c = 6, d = 18, etc. But as we saw, the latter gives higher sum.

Therefore, I think the answer is 16. However, I need to check if there is a possible mistake in my initial assumption. For example, when I derived the area formula, I assumed that the area is (d - c)^2 / |a - b|. Let me verify this with another example.

Take a = 1, b = 2, c = 1, d = 3. Then, the first parallelogram area should be (3 - 1)^2 / |1 - 2| = 4 / 1 = 4. Let me compute it with coordinates.

Lines y = x + 1, y = x + 3, y = 2x + 1, y = 2x + 3.

Vertices:

Intersection of y = x + 1 and y = 2x + 1: x + 1 = 2x + 1 => x = 0, y = 1. Point (0, 1).

Intersection of y = x + 1 and y = 2x + 3: x + 1 = 2x + 3 => -x = 2 => x = -2, y = -2 + 1 = -1. Point (-2, -1).

Intersection of y = x + 3 and y = 2x + 1: x + 3 = 2x + 1 => -x = -2 => x = 2, y = 2 + 3 = 5. Point (2, 5).

Intersection of y = x + 3 and y = 2x + 3: x + 3 = 2x + 3 => x = 0, y = 3. Point (0, 3).

So vertices are (0, 1), (-2, -1), (0, 3), (2, 5). Applying shoelace formula:

Order: (0, 1), (-2, -1), (0, 3), (2, 5), back to (0, 1).

Compute terms:

First term: 0*(-1) - (-2)*1 = 0 + 2 = 2

Second term: -2*3 - 0*(-1) = -6 - 0 = -6

Third term: 0*5 - 2*3 = 0 - 6 = -6

Fourth term: 2*1 - 0*5 = 2 - 0 = 2

Sum: 2 - 6 - 6 + 2 = -8

Area = 1/2 * | -8 | = 4. Correct. Which matches the formula. Therefore, the formula is correct.

Therefore, my formula is correct. Therefore, the answer is 16.

But wait, the problem says "the smallest possible value of a + b + c + d". However, I need to check if there are other solutions with smaller sum. For example, if there is a case where a - b = 2, and c and d are smaller. But as we saw, c and d must satisfy c + d = 12 and d - c = 6, leading to c = 3, d = 9. Therefore, no smaller c and d possible. Therefore, 16 is indeed the minimal.

Therefore, I think the answer is 16. However, I need to check if the problem has any constraints that I missed. For example, the problem says that the lines are y = ax + c, y = ax + d, etc. Are there any restrictions on the lines? For example, they must be distinct, which they are since c ≠ d and a ≠ b. Also, the problem says that a, b, c, d are positive integers. Which they are in our case.

Therefore, I think the answer is 16. However, I recall that in some cases, the minimal sum might be higher. But according to all calculations, 16 is valid. However, I need to check if there is a possible mistake in the problem statement or my interpretation.

Wait, the problem says "the parallelogram bounded by the lines y=ax+c, y=ax+d, y=bx+c, and y=bx+d has area 18". In our case, with a = 3, b = 1, c = 3, d = 9, the lines are y = 3x + 3, y = 3x + 9, y = x + 3, y = x + 9. These are four lines forming a parallelogram. Similarly for the other case. Therefore, all conditions are satisfied.

Therefore, I think the answer is 16. However, I need to check if there is a possible mistake in my calculation of the area formula. For example, if I had a different formula. But since I verified with coordinates and shoelace formula, it's correct.

Therefore, I think the answer is 16. However, I need to check if there is a possible smaller sum. For example, if there is a case where a - b = 2, and c = 3, d = 9, but with a = 1, b = 3, which gives the same sum. Therefore, no.

Therefore, I think the answer is 16. However, I recall that in some Olympiad problems, sometimes there are hidden constraints or alternative solutions. But given the time I've spent and multiple checks, I think 16 is correct.

**Final Answer**
\boxed{16}

To solve the problem, we are given two parallelograms formed by lines with slopes $ a $ and $ b $, and constants $ c $ and $ d $. The first parallelogram is bounded by the lines:

- $ y = ax + c $
- $ y = ax + d $
- $ y = bx + c $
- $ y = bx + d $

The second parallelogram is bounded by:

- $ y = ax + c $
- $ y = ax - d $
- $ y = bx + c $
- $ y = bx - d $

We are told that the area of the first parallelogram is 18, and the area of the second is 72. We are to find the **smallest possible value of** $ a + b + c + d $, where $ a, b, c, d $ are **positive integers**.

---

### Step 1: Understanding the Area of a Parallelogram Formed by Lines

The area of a parallelogram formed by two pairs of parallel lines can be computed using the formula:

$$
\text{Area} = \frac{(c + d)^2}{|a - b|} \quad \text{or} \quad \frac{(d - c)^2}{|a - b|}
$$

depending on the configuration of the lines. In our case:

- For the **first parallelogram**, the lines are $ y = ax + c $, $ y = ax + d $, $ y = bx + c $, $ y = bx + d $, so the area is:

$$
\text{Area}_1 = \frac{(d - c)^2}{|a - b|} = 18
$$

- For the **second parallelogram**, the lines are $ y = ax + c $, $ y = ax - d $, $ y = bx + c $, $ y = bx - d $, so the area is:

$$
\text{Area}_2 = \frac{(c + d)^2}{|a - b|} = 72
$$

---

### Step 2: Relating the Two Equations

Let’s denote $ k = |a - b| $. Then we have:

$$
\frac{(d - c)^2}{k} = 18 \quad \text{and} \quad \frac{(c + d)^2}{k} = 72
$$

Dividing the second equation by the first:

$$
\frac{(c + d)^2}{(d - c)^2} = \frac{72}{18} = 4
\Rightarrow \left( \frac{c + d}{d - c} \right)^2 = 4
\Rightarrow \frac{c + d}{d - c} = 2
$$

This gives us the equation:

$$
c + d = 2(d - c)
\Rightarrow c + d = 2d - 2c
\Rightarrow 3c = d
$$

So, $ d = 3c $.

---

### Step 3: Substituting Back

Now substitute $ d = 3c $ into the first area equation:

$$
\frac{(d - c)^2}{k} = 18
\Rightarrow \frac{(3c - c)^2}{k} = 18
\Rightarrow \frac{(2c)^2}{k} = 18
\Rightarrow \frac{4c^2}{k} = 18
\Rightarrow k = \frac{2c^2}{9}
$$

Since $ k = |a - b| $, and $ a, b $ are integers, $ k $ must be a positive integer. Therefore, $ 2c^2 $ must be divisible by 9. Since 2 and 9 are coprime, $ c^2 $ must be divisible by 9, so $ c $ must be divisible by 3.

Let $ c = 3k $, then $ d = 9k $, and:

$$
k = \frac{2(3k)^2}{9} = \frac{2 \cdot 9k^2}{9} = 2k^2
$$

So, $ a - b = 2k^2 $, and we can write $ a = b + 2k^2 $.

---

### Step 4: Minimizing the Sum $ a + b + c + d $

We now express the total sum in terms of $ k $ and $ b $:

$$
a + b + c + d = (b + 2k^2) + b + 3k + 9k = 2b + 2k^2 + 12k
$$

To minimize this, we choose the smallest possible values of $ b $ and $ k $, both positive integers.

Let’s try $ k = 1 $, $ b = 1 $:

- $ c = 3k = 3 $
- $ d = 9k = 9 $
- $ a = b + 2k^2 = 1 + 2 = 3 $

So the values are:

- $ a = 3 $
- $ b = 1 $
- $ c = 3 $
- $ d = 9 $

Check the areas:

- First area: $ \frac{(d - c)^2}{a - b} = \frac{(9 - 3)^2}{3 - 1} = \frac{36}{2} = 18 $
- Second area: $ \frac{(c + d)^2}{a - b} = \frac{(3 + 9)^2}{2} = \frac{144}{2} = 72 $

Both match the given areas.

Now compute the sum:

$$
a + b + c + d = 3 + 1 + 3 + 9 = \boxed{16}
$$

---

### Final Answer

$$
\boxed{16}
$$

Solution:
Two vertices of the first parallelogram are at $(0,c)$ and $(0,d)$.

[asy]
unitsize(0.5 cm);

pair P, Q, R, S;

P = (0,9);
Q = (3,12);
R = (0,3);
S = (-3,0);

draw(interp(P,Q,-0.4)--interp(P,Q,1.4));
draw(interp(R,S,-0.4)--interp(R,S,1.4));
draw(interp(P,S,-0.2)--interp(P,S,1.2));
draw(interp(Q,R,-0.2)--interp(Q,R,1.2));

label("$y = ax + c$", interp(S,R,1.4), E);
label("$y = ax + d$", interp(P,Q,1.4), E);
label("$y = bx + c$", interp(Q,R,1.2), SE);
label("$y = bx + d$", interp(P,S,1.2), SE);

dot("$(0,c)$", R, SE);
dot("$(0,d)$", P, NW);
[/asy]

The $x$-coordinates of the other two vertices satisfy $ax+c=bx+d$ and $ax+d=bx+c$, so the $x$-coordinates are $\pm(c-d)/(b-a)$. Thus the parallelogram is composed of two triangles, each of which has area \[
9=\frac{1}{2} \cdot |c-d| \cdot \left|\frac{c-d}{b-a}\right|.
\]It follows that $(c-d)^2=18|b-a|$.

By a similar argument using the second parallelogram, $(c+d)^2=72|b-a|$. Subtracting the first equation from the second yields $4cd=54|b-a|$, so $2cd = 27|b-a|$. Thus $|b-a|$ is even, and $a+b$ is minimized when $\{a,b\}=\{1,3\}$. Also, $cd$ is a multiple of 27, and $c+d$ is minimized when $\{c,d\}=\{3,9\}$. Hence the smallest possible value of $a+b+c+d$ is $1+3+3+9=\boxed{16}$. Note that the required conditions are satisfied when $(a,b,c,d)=(1,3,3,9)$.