Explanation:
Okay, so I need to find the smallest possible value of a + b + c + d where a, b, c, d are positive integers. The problem gives two parallelograms with areas 18 and 72, defined by different sets of lines. Let me try to visualize and understand the problem step by step.

First, let me recall that the area of a parallelogram can be calculated if we know the base and the height, but since these parallelograms are defined by lines, maybe there's a formula related to the distance between lines and the angle between them? Hmm.

The first parallelogram is bounded by y = ax + c, y = ax + d, y = bx + c, and y = bx + d. Similarly, the second one is bounded by y = ax + c, y = ax - d, y = bx + c, and y = bx - d. The areas are given as 18 and 72 respectively.

Since these are parallelograms formed by two pairs of parallel lines, each pair of lines with the same slope (a and b). The distance between the lines y = ax + c and y = ax + d should be the same for both pairs, but since the second parallelogram has different constants (c and -d), maybe the distances are related?

Wait, let me think. For two parallel lines y = mx + c1 and y = mx + c2, the distance between them is |c1 - c2| / sqrt(m² + 1). So for the first parallelogram, the vertical distances between the lines would be |c - d| / sqrt(a² + 1) and |c - d| / sqrt(b² + 1)? Wait, no, actually, since both pairs of lines are parallel. The distance between 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). But since it's a parallelogram, the area would be the product of these two distances multiplied by the sine of the angle between the lines? Wait, no, actually, if we have two pairs of lines, the area of the parallelogram is the product of the distances between the two pairs times the sine of the angle between the two lines? Wait, maybe?

Alternatively, maybe the area can be calculated by the distance between one pair of lines multiplied by the distance between the other pair divided by sin(theta), where theta is the angle between the lines? Wait, I need to recall the formula for area of a parallelogram formed by two pairs of lines.

Alternatively, think of the vectors. If we have two sets of lines, the sides of the parallelogram can be determined by the direction vectors of the lines. Let me try this approach.

Let me take the first parallelogram: lines y = ax + c, y = ax + d (both with slope a), and lines y = bx + c, y = bx + d (both with slope b). The intersection points of these lines will form the vertices of the parallelogram. Let me find the coordinates of these vertices.

First, find intersection of y = ax + c and y = bx + c. Setting ax + c = bx + c => ax = bx => (a - b)x = 0. Since a ≠ b (otherwise, they would be parallel lines and not form a parallelogram), so x = 0. Then y = a*0 + c = c. So one vertex is (0, c).

Second, intersection of y = ax + c and y = bx + d. Set ax + c = bx + d => (a - b)x = d - c => x = (d - c)/(a - b). Then y = a*( (d - c)/(a - b) ) + c = [a(d - c) + c(a - b)] / (a - b) = [ad - ac + ac - bc]/(a - b) = (ad - bc)/(a - b). Let me write that as y-coordinate.

Third, intersection of y = ax + d and y = bx + c. Similarly, ax + d = bx + c => (a - b)x = c - d => x = (c - d)/(a - b). Then y-coordinate: same as above, but with sign switched? Let's compute:

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).

Fourth, intersection of y = ax + d and y = bx + d. Setting ax + d = bx + d => (a - b)x = 0 => x = 0, then y = d. So fourth vertex is (0, d).

Wait, so the four vertices are (0, c), [(d - c)/(a - b), (ad - bc)/(a - b)], [(c - d)/(a - b), (ac - bd)/(a - b)], and (0, d). Wait, but [(c - d)/(a - b), (ac - bd)/(a - b)] is equivalent to [-(d - c)/(a - b), (ac - bd)/(a - b)]. Let me check if this makes sense.

Alternatively, maybe I can think of vectors between these points to compute the area. But perhaps there's a formula for the area of a parallelogram formed by two pairs of lines. Let me recall that.

In general, if you have two pairs of lines: one pair with slope m1: y = m1x + c1 and y = m1x + c2, and another pair with slope m2: y = m2x + d1 and y = m2x + d2. Then the area of the parallelogram formed by these lines is |(c2 - c1)(d2 - d1)| / |m1 - m2|. Wait, no, that seems too simple. Wait, maybe?

Wait, actually, the distance between the first pair of lines is |c1 - c2| / sqrt(m1² + 1), and the distance between the second pair is |d1 - d2| / sqrt(m2² + 1). But the area is the product of these distances divided by the sine of the angle between the lines? Wait, no. Wait, in general, if you have two vectors, the area is |v × w|. So maybe we need to find vectors along the sides of the parallelogram and take the cross product.

Let me try that.

For the first parallelogram, with lines y = ax + c, y = ax + d, y = bx + c, y = bx + d. Let me find two vectors that form the sides of the parallelogram.

Take the vertex (0, c) and move along the direction of the line y = ax + c, which has direction vector (1, a). Similarly, moving along y = bx + c, direction vector (1, b). Wait, but actually, the sides of the parallelogram are the directions of the lines? Wait, maybe not. Let me think.

Alternatively, the vectors representing the sides of the parallelogram can be found by taking the difference between two adjacent vertices. For example, from (0, c) to the intersection point [(d - c)/(a - b), (ad - bc)/(a - b)], which I can call point A. Let me compute vector OA - vector OB.

Wait, maybe it's easier to compute the vectors. Let me take two adjacent sides of the parallelogram. For example, from (0, c) to [(d - c)/(a - b), (ad - bc)/(a - b)] and from (0, c) to [(c - d)/(a - b), (ac - bd)/(a - b)]. Wait, but maybe I need to find two vectors that are along the sides. Alternatively, since the sides are between the lines, perhaps the vectors can be determined by the direction vectors of the lines?

Alternatively, maybe we can use the formula for the area of a parallelogram defined by two lines. Wait, I found a resource once that mentioned that if you have two sets of parallel lines, the area can be calculated as (distance between first pair) * (distance between second pair) / sin(theta), where theta is the angle between the two sets of lines. Wait, but why divided by sin(theta)?

Wait, suppose you have two vectors forming adjacent sides of a parallelogram. The area is the magnitude of their cross product. If the two lines are at an angle theta, then the cross product would involve sin(theta). Let me think.

Alternatively, let me think of the area as the base times height. Suppose one pair of lines defines the base, and the other pair defines the height. Wait, but the height would depend on the angle between the lines.

But maybe there's a formula here. Let me try to recall. Suppose we have two lines with slopes m1 and m2. The angle theta between them is given by tan(theta) = |(m2 - m1)/(1 + m1m2)|. Then, if I can find the distances between the lines and multiply them by sin(theta), that would give the area?

Wait, if we take the distance between the first pair of lines, which is D1, and the distance between the second pair, D2, then the area of the parallelogram should be D1 * D2 / sin(theta)? Wait, no. Wait, if theta is the angle between the lines, then if you have a base of length D1 and a height corresponding to the other side, which is D2 / sin(theta)?

This is getting confusing. Maybe I need to use coordinates.

Alternatively, since I have the coordinates of the vertices, maybe I can use the shoelace formula to compute the area. Let me try that.

First, for the first parallelogram. Let me denote the four vertices:

Vertex 1: (0, c)

Vertex 2: [(d - c)/(a - b), (ad - bc)/(a - b)]

Vertex 3: [(c - d)/(a - b), (ac - bd)/(a - b)]

Vertex 4: (0, d)

Wait, but if I connect these points, is that a parallelogram? Let me check if opposite sides are equal and parallel. Let me compute the vectors between these points.

Vector from Vertex 1 to Vertex 2: [(d - c)/(a - b) - 0, (ad - bc)/(a - b) - c] = [(d - c)/(a - b), (ad - bc - c(a - b))/(a - b)] = [(d - c)/(a - b), (ad - bc - ac + bc)/(a - b)] = [(d - c)/(a - b), (ad - ac)/(a - b)] = [(d - c)/(a - b), a(d - c)/(a - b)] = (d - c)/(a - b) * (1, a)

Similarly, vector from Vertex 4 to Vertex 3: [(c - d)/(a - b) - 0, (ac - bd)/(a - b) - d] = [(c - d)/(a - b), (ac - bd - d(a - b))/(a - b)] = [(c - d)/(a - b), (ac - bd - ad + bd)/(a - b)] = [(c - d)/(a - b), (ac - ad)/(a - b)] = [(c - d)/(a - b), a(c - d)/(a - b)] = (c - d)/(a - b)*(1, a) = -(d - c)/(a - b)*(1, a)

Which is exactly the negative of the vector from Vertex 1 to Vertex 2, so that makes sense for a parallelogram. Similarly, vectors between other sides.

Similarly, vector from Vertex 1 to Vertex 4 is (0 - 0, d - c) = (0, d - c). Then, vector from Vertex 2 to Vertex 3: [(c - d)/(a - b) - (d - c)/(a - b), (ac - bd)/(a - b) - (ad - bc)/(a - b)] = [ (c - d - d + c)/(a - b), (ac - bd - ad + bc)/(a - b) ] = [ (2c - 2d)/(a - b), (ac - ad + bc - bd)/(a - b) ] = [ 2(c - d)/(a - b), a(c - d) + b(c - d) / (a - b) ] = [ 2(c - d)/(a - b), (a + b)(c - d)/(a - b) ] = (c - d)/(a - b)*(2, a + b)

Wait, but the vector from Vertex 1 to Vertex 4 is (0, d - c). Let me check if this is parallel to the vector from Vertex 2 to Vertex 3. The vector from Vertex 1 to Vertex 4 is (0, d - c), and the vector from Vertex 2 to Vertex 3 is (c - d)/(a - b)*(2, a + b). For them to be parallel, there must be a scalar multiple. Let me see:

(0, d - c) = k*(2(c - d)/(a - b), (a + b)(c - d)/(a - b))

But (d - c) = -(c - d), so:

(0, - (c - d)) = k*(2(c - d)/(a - b), (a + b)(c - d)/(a - b))

So, if we let k = - (a - b)/2, then:

Left side: (0, - (c - d))

Right side: - (a - b)/2 * [ 2(c - d)/(a - b), (a + b)(c - d)/(a - b) ) ] = - (a - b)/2 * 2(c - d)/(a - b) , - (a - b)/2 * (a + b)(c - d)/(a - b) ) = - (c - d), - (a + b)(c - d)/2 )

Wait, that doesn't match the left side. So maybe I made a mistake here. Wait, the vector from Vertex 1 to Vertex 4 is (0, d - c). The vector from Vertex 2 to Vertex 3 is something else. Maybe I need to find another vector.

Alternatively, maybe I can use the shoelace formula. Let me list the coordinates of the vertices in order. Let me suppose they are ordered as follows: (0, c), [(d - c)/(a - b), (ad - bc)/(a - b)], (0, d), [(c - d)/(a - b), (ac - bd)/(a - b)]? Wait, not sure of the order. Maybe it's better to go around the parallelogram.

Alternatively, maybe I can compute the area using vectors. The area is |(v1 × v2)|, where v1 and v2 are vectors representing adjacent sides.

Earlier, I found that one side vector is (d - c)/(a - b)*(1, a), and another side vector can be found from (0, d - c). Wait, but the vector from Vertex 1 (0, c) to Vertex 4 (0, d) is (0, d - c). So if one vector is (d - c)/(a - b)*(1, a) and another vector is (0, d - c), then the cross product (in 2D, the magnitude is |v1_x v2_y - v1_y v2_x|).

Let me compute that. Let me denote vector u = [(d - c)/(a - b), a(d - c)/(a - b)] and vector v = [0, d - c]

Then cross product magnitude |u_x v_y - u_y v_x| = | [(d - c)/(a - b)*(d - c) - a(d - c)/(a - b)*0 ] | = | (d - c)^2 / (a - b) |. Therefore, the area of the parallelogram is |(d - c)^2 / (a - b)|. Wait, but the area is given as 18.

Wait, so according to this, Area1 = |(d - c)^2 / (a - b)| = 18. Similarly, for the second parallelogram, which is bounded by y = ax + c, y = ax - d, y = bx + c, y = bx - d. Let me compute its area similarly.

So for the second parallelogram, the constants are c and -d instead of c and d. Following the same logic, the distance between the lines y = ax + c and y = ax - d is |c - (-d)| / sqrt(a² + 1) = |c + d| / sqrt(a² + 1). But wait, but according to the previous calculation, the area is |(difference in constants)^2 / (a - b)|. Wait, in the first case, the difference between the constants was (d - c), but here, the difference would be (-d - c) = -(c + d). Then, the area would be |(- (c + d))² / (a - b)| = |(c + d)^2 / (a - b)| = 72.

Wait, but according to the previous calculation, for the first parallelogram, the area was |(d - c)^2 / (a - b)| = 18, and for the second one, it's |(c + d)^2 / (a - b)| = 72. Therefore, we have two equations:

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

2. (c + d)^2 / |a - b| = 72

Since all variables are positive integers, and a and b are positive integers, but we don't know if a > b or b > a. However, since the denominators are absolute values, we can assume that |a - b| is positive, so the denominators are same for both equations. Therefore, we can denote k = |a - b|, so that:

(d - c)^2 / k = 18

(c + d)^2 / k = 72

Therefore, if we take the ratio of the second equation to the first, we get [(c + d)^2 / k] / [(d - c)^2 / k] = 72 / 18 = 4. Therefore, [(c + d)^2] / [(d - c)^2] = 4 => (c + d)/(d - c) = 2 (since c + d and d - c are positive? Wait, since c and d are positive integers, but we need to check if d - c is positive. If d - c is positive, then (c + d)/(d - c) = 2 => c + d = 2(d - c) => c + d = 2d - 2c => c + d - 2d + 2c = 0 => 3c - d = 0 => d = 3c. Alternatively, if d - c is negative, then |d - c| is positive. However, since (d - c)^2 is positive, and the equations are in terms of absolute value, but when we took the ratio, we assumed (d - c)^2 is positive, so regardless, the ratio gives [(c + d)^2]/[(d - c)^2] = 4. Therefore, taking square roots, (c + d)/|d - c| = 2. Since c and d are positive integers, let's suppose d > c (so that d - c is positive). Therefore, (c + d)/(d - c) = 2 => c + d = 2d - 2c => 3c = d.

Alternatively, if d < c, then |d - c| = c - d, so (c + d)/(c - d) = 2 => c + d = 2c - 2d => 3d = c. But since c and d are positive integers, both cases are possible. However, since the problem states that a, b, c, d are positive integers, and we need to find the minimal sum, we need to consider both cases.

So, first case: d = 3c. Second case: c = 3d. Let me check both possibilities.

Let me consider case 1: d = 3c.

Substituting back into the first equation:

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

Similarly, from the second equation:

(c + d)^2 / k = 72 => (c + 3c)^2 / k = 72 => (4c)^2 / k = 72 => 16c² / k = 72 => 2c² / k = 9. Same as the first equation. So consistent.

Therefore, k = 2c² / 9. But k is |a - b|, which must be a positive integer. Therefore, 2c² must be divisible by 9, so c² must be divisible by 9/ gcd(2, 9) = 9, since 2 and 9 are coprime. Therefore, c² must be divisible by 9 => c must be divisible by 3. Let me write c = 3m, where m is a positive integer. Then, d = 3c = 9m. Then, k = 2*(3m)^2 / 9 = 2*9m² / 9 = 2m². Therefore, |a - b| = 2m². Since a and b are positive integers, we can let a = b + 2m² or b = a + 2m². But since a and b are positive integers, both possibilities are allowed. However, to minimize a + b + c + d, we need to choose m as small as possible.

Similarly, in case 2: c = 3d. Then, substituting into first equation:

(d - c)^2 / k = 18 => (d - 3d)^2 / k = (-2d)^2 / k = 4d² / k = 18 => 2d² / k = 9 => k = 2d² / 9. Similarly, second equation:

(c + d)^2 / k = (3d + d)^2 / k = (4d)^2 / k = 16d² / k = 72 => 2d² / k = 9, same as before. Therefore, k = 2d² / 9. Then, since c = 3d, and c must be divisible by 3, which it is. Similarly, d must be such that 2d² is divisible by 9. Therefore, d must be divisible by 3. Let me set d = 3n, then c = 3d = 9n. Then k = 2*(3n)^2 / 9 = 2*9n² /9 = 2n². Therefore, |a - b| = 2n². Again, same as case 1.

Therefore, in both cases, the structure is similar: either c = 3m and d = 9m, or c = 9n and d = 3n, with |a - b| = 2m² or 2n². Therefore, to minimize a + b + c + d, we need to minimize m (or n) and choose between cases.

But note that in case 1, c = 3m, d = 9m, and in case 2, c = 9n, d = 3n. Let me compare. For case 1, the sum c + d = 3m + 9m = 12m. For case 2, c + d = 9n + 3n = 12n. So they are the same for same m or n. Therefore, which case is better? Well, since we need to minimize the total sum a + b + c + d. Let's consider case 1: c = 3m, d = 9m. Then, |a - b| = 2m². So to minimize a + b, given that |a - b| = 2m². Let me let a and b be such that a - b = 2m² or b - a = 2m². To minimize a + b, we can set one variable as small as possible. For example, let me assume a > b, so a = b + 2m². Then, a + b = 2b + 2m². To minimize this, set b as small as possible. Since b is a positive integer, minimal b is 1. Therefore, a = 1 + 2m², and then a + b = 2 + 2m².

Similarly, in case 2, c = 9n, d = 3n. Then |a - b| = 2n². Similarly, minimal a + b would be 2 + 2n². Therefore, in both cases, the sum a + b is 2 + 2m² or 2 + 2n². However, c + d is 12m in case 1 or 12n in case 2. Therefore, total sum is (2 + 2m²) + 12m or (2 + 2n²) + 12n. So, which gives a smaller total? Let's see for minimal m and n.

Since m and n are positive integers, starting with m = 1, n = 1.

Case 1, m = 1: total sum = 2 + 2(1)^2 + 12*1 = 2 + 2 + 12 = 16

Case 2, n = 1: total sum = 2 + 2(1)^2 + 12*1 = same 16

But we need to check if this is possible. Let me check if a, b, c, d are positive integers.

For case 1: m = 1, c = 3*1 = 3, d = 9*1 = 9. Then |a - b| = 2*(1)^2 = 2. Let me choose b = 1, then a = b + 2 = 3. Therefore, a = 3, b = 1. All positive integers. Then check if the given areas hold. Let's verify.

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. Therefore, this works. So total sum is a + b + c + d = 3 + 1 + 3 + 9 = 16. Wait, 3 + 1 is 4, 3 + 9 is 12, total 16. However, let me check case 2 with n = 1. c = 9*1 = 9, d = 3*1 = 3. Then |a - b| = 2*(1)^2 = 2. Let me choose b = 1, then a = 1 + 2 = 3. Then sum a + b + c + d = 3 + 1 + 9 + 3 = same 16. So same result.

But wait, the problem says that a, b, c, d are positive integers. However, in case 1, we have a = 3, b = 1, c = 3, d = 9. But the lines in the first parallelogram are y = ax + c = 3x + 3, y = 3x + 9, y = bx + c = 1x + 3, y = 1x + 9. The second parallelogram is y = 3x + 3, y = 3x - 9, y = x + 3, y = x - 9. Wait, but the second parallelogram is bounded by y = ax + c, y = ax - d, y = bx + c, y = bx - d. Here, ax - d would be 3x - 9, which is valid. Similarly, bx - d is x - 9, which is okay. So, that seems okay. So, total sum is 16. However, the question is asking for the smallest possible value of a + b + c + d. But 16 seems low. Wait, but maybe there's a way to get a smaller sum?

Wait, maybe m or n can be 0? But m and n are positive integers, so starting at 1. But maybe there's another case?

Wait, but in the ratio we had (c + d)/(d - c) = 2 or similar, leading to d = 3c or c = 3d. But is there another possibility? For example, if d - c is negative, then |d - c| = c - d. But when taking the ratio, we had [(c + d)^2]/[(d - c)^2] = 4, so [(c + d)/|d - c|]^2 = 4. Therefore, (c + d)/|d - c| = 2 or -2. But since c and d are positive integers, c + d is positive, and |d - c| is positive. Therefore, (c + d)/|d - c| = 2. Therefore, only that case.

So the only possibilities are d = 3c or c = 3d. So we need to check if there are other solutions. But in both cases, when m = 1, we get the total sum as 16. However, maybe there's a case where |a - b| is smaller? Let me check.

Wait, for example, if in case 1: d = 3c, and k = |a - b| = 2c² / 9. Since k must be integer, 2c² must be divisible by 9. Therefore, c must be divisible by 3, which we already considered. So if c = 3m, then k = 2*(3m)^2 / 9 = 2*9m² / 9 = 2m². Therefore, k = 2m². Similarly for case 2. Therefore, with m = 1, k = 2, which is integer. So that's okay.

But is there another way? Suppose that c and d are such that (c + d)^2 / (d - c)^2 = 4, but with different relationships between c and d. For example, if d - c = something else? Wait, but algebraically we have that [(c + d)/(d - c)] = 2, which gives d = 3c or c = 3d. So no other possibilities. Hence, the minimal m or n is 1, leading to total sum 16. However, the answer might be different. Wait, wait, but maybe there is another approach where the area is calculated differently?

Wait, earlier I derived the area as |(d - c)^2 / (a - b)|. Wait, but I need to check that again. Let me go back. When I calculated the cross product earlier, I had vectors u = [(d - c)/(a - b), a(d - c)/(a - b)] and v = [0, d - c]. Then cross product was |u_x v_y - u_y v_x| = | [(d - c)/(a - b)*(d - c) - a(d - c)/(a - b)*0 ]| = | (d - c)^2 / (a - b) |. Therefore, the area is |(d - c)^2 / (a - b)|. But is this correct? Let me check with a simple example.

Suppose a = 1, b = 0, c = 1, d = 2. Then lines y = x + 1, y = x + 2, y = 0x + 1, y = 0x + 2. The parallelogram is between these lines. The area should be |(2 - 1)^2 / (1 - 0)| = 1/1 = 1. Let me compute it manually. The vertices are (0, 1), intersection of y = x + 1 and y = 0x + 2: x + 1 = 2 => x = 1, y = 2. So (1, 2). Similarly, intersection of y = x + 2 and y = 0x + 1: x + 2 = 1 => x = -1, y = 1. So (-1, 1). And intersection of y = x + 1 and y = 0x + 1 is (0, 1). Wait, so vertices are (0, 1), (1, 2), (0, 2), (-1, 1). Using shoelace formula:

Coordinates: (0,1), (1,2), (0,2), (-1,1), back to (0,1).

Compute area: 1/2 |(0*2 + 1*2 + 0*1 + (-1)*1) - (1*1 + 2*0 + 2*(-1) + 1*0)|

Wait, let me use the shoelace formula step by step:

List the coordinates:

(0,1), (1,2), (0,2), (-1,1), (0,1)

Compute sum of x_i y_{i+1}:

0*2 + 1*2 + 0*1 + (-1)*1 = 0 + 2 + 0 -1 = 1

Compute sum of y_i x_{i+1}:

1*1 + 2*0 + 2*(-1) + 1*0 = 1 + 0 - 2 + 0 = -1

Subtract: 1 - (-1) = 2. Take absolute value and divide by 2: |2| / 2 = 1. Which matches the formula result. Therefore, my formula seems correct. Therefore, for the first case, area is |(d - c)^2 / (a - b)|. Therefore, previous calculations are correct.

Therefore, in our first case, the minimal total sum is 16. However, maybe there's a mistake here? Let me check if the problem says that the lines are "bounded by" the given lines. Is there a possibility that when we take a and b, they might need to satisfy some other condition? For example, if a = b, then the lines would be parallel, but since the problem says they form a parallelogram, a ≠ b must hold. Which they do in our case, since |a - b| = 2m² ≠ 0.

But let me check if a and b are positive integers. In our case, for m = 1, a = 3, b = 1, which are positive integers. Similarly, c = 3, d = 9. So all are positive integers. Therefore, 16 is possible. However, the problem says "the smallest possible value of a + b + c + d". Is 16 indeed the minimal? Or maybe there is a smaller sum with different relationships?

Wait, but maybe there is a case where (c + d)^2 / (d - c)^2 = 4 is satisfied with different c and d? For example, if c + d = 2(d - c), which gives d = 3c, or c + d = -2(d - c), but since all are positive, the second equation would lead to c + d = -2d + 2c => 3d = c. So same as before.

Alternatively, could there be cases where the ratio is 4 but with different relations? For instance, if (c + d)^2 = 4(d - c)^2. Then sqrt of both sides: c + d = ±2(d - c). Since all are positive, c + d = 2(d - c) => d = 3c or c + d = -2(d - c) => c + d = -2d + 2c => 3d = c. So again same results. Therefore, no other possibilities. Therefore, minimal is 16? But let me check.

Wait, but let me check with m = 1 gives sum 16. However, let me check if there is a case with m = 1, but different a and b. For example, if a and b are such that |a - b| = 2, but different from a = 3, b = 1. For example, a = 2, b = 0. But b must be positive integer, so b = 0 is invalid. Similarly, a = 4, b = 2. Then a + b = 6. But c = 3, d = 9, so total sum would be 4 + 2 + 3 + 9 = 18, which is larger than 16. Hence, the minimal in case 1 is when a = 3, b = 1, giving sum 16.

But maybe there are other possibilities where the ratio is 4, but with different values?

Alternatively, perhaps my assumption that k = |a - b| is the same for both equations is incorrect? Wait, no, because both areas have the same denominator, which is |a - b|. Therefore, if we have [(d - c)^2]/|a - b| = 18 and [(c + d)^2]/|a - b| = 72, then dividing gives 4, leading to the same relation between c and d. Therefore, I think this is correct.

But let me check with different numbers. Suppose I take c = 3, d = 9, then (c + d) = 12, (d - c) = 6. 12^2 / 6^2 = 144 / 36 = 4. Correct. Then, if |a - b| = 2, then first area is 36 / 2 = 18, second is 144 / 2 = 72. Correct.

But maybe if there are other values of c and d where [(c + d)^2]/[(d - c)^2] = 4 but with smaller c and d? For example, if c = 1, d = 3. Then (1 + 3)^2 / (3 - 1)^2 = 16 / 4 = 4. Then, this would satisfy the ratio. Wait! Wait, this is case 2, where c = 3d? Wait no, here c = 1, d = 3. So d = 3c? No, d = 3*1 = 3, so d = 3c. Hence, case 1. Wait, if c = 1, d = 3, then |a - b| = [ (d - c)^2 ] / 18 = (2)^2 / 18 = 4 / 18 = 2/9. Which is not integer. Therefore, invalid. Therefore, even though the ratio is 4, the denominator |a - b| would not be integer. Therefore, this is impossible. Hence, why in our previous reasoning, we needed c to be multiple of 3.

So for example, if c = 3m, d = 9m, then (d - c) = 6m, (c + d) = 12m, then [(c + d)^2]/[(d - c)^2] = (144m²)/(36m²) = 4. Then |a - b| = (d - c)^2 / 18 = (36m²)/18 = 2m². Therefore, for m = 1, we get |a - b| = 2*1 = 2, which is integer. If m = 1, then c = 3, d = 9. If m = 2, c = 6, d = 18, |a - b| = 8, etc. Similarly for other m's.

Therefore, for m = 1, we get minimal sum. Therefore, 16?

But wait, let me check the problem statement again. It says "the parallelogram bounded by the lines y=ax+c, y=ax+d, y=bx+c, and y=bx+d has area 18". Similarly for the other. If we take a = 3, b = 1, c = 3, d = 9, then the first parallelogram is between y = 3x + 3, y = 3x + 9, y = x + 3, y = x + 9. Let me confirm the area with shoelace formula.

Vertices:

Intersection of y=3x+3 and y=x+3: 3x + 3 = x + 3 ⇒ 2x = 0 ⇒ x=0, y=3. So (0, 3).

Intersection of y=3x+3 and y=x+9: 3x + 3 = x + 9 ⇒ 2x = 6 ⇒ x = 3, y = 3*3 + 3 = 12. So (3, 12).

Intersection of y=3x+9 and y=x+9: 3x + 9 = x + 9 ⇒ 2x = 0 ⇒ x=0, y=9. So (0, 9).

Intersection of y=3x+9 and y=x+3: 3x + 9 = x + 3 ⇒ 2x = -6 ⇒ x = -3, y = -3 + 3 = 0. So (-3, 0).

Wait, but hold on, these points may not be in order. Let me list all four vertices:

(0,3), (3,12), (0,9), (-3,0). Wait, connecting these? Let me check coordinates again:

Wait, actually, the four lines are:

1. y = 3x + 3

2. y = 3x + 9

3. y = x + 3

4. y = x + 9

So intersections:

1 & 3: (0,3)

1 & 4: 3x + 3 = x + 9 ⇒ 2x = 6 ⇒ x = 3, y = 3*3 + 3 = 12 ⇒ (3,12)

2 & 3: 3x + 9 = x + 3 ⇒ 2x = -6 ⇒ x = -3, y = -3 + 3 = 0 ⇒ (-3, 0)

2 & 4: 3x + 9 = x + 9 ⇒ 2x = 0 ⇒ x = 0, y = 0 + 9 = 9 ⇒ (0,9)

So the four vertices are (0,3), (3,12), (0,9), (-3, 0). Let me order them properly for shoelace.

Let me arrange them in order. Let me think of the parallelogram. Starting at (0,3), moving along y=3x + 3 to (3,12)? Wait, no. Actually, the sides are between the lines. Maybe the correct order is (0,3), (3,12), (0,9), (-3,0), back to (0,3). Let me apply the shoelace formula:

List the coordinates in order:

(0, 3), (3, 12), (0, 9), (-3, 0), (0, 3)

Compute shoelace sum:

Sum1 = (0*12 + 3*9 + 0*0 + (-3)*3) = 0 + 27 + 0 -9 = 18

Sum2 = (3*3 + 12*0 + 9*(-3) + 0*0) = 9 + 0 -27 + 0 = -18

Area = 1/2 |Sum1 - Sum2| = 1/2 |18 - (-18)| = 1/2 * 36 = 18. Correct. So that works.

For the second parallelogram with lines y=3x+3, y=3x-9, y=x+3, y=x-9.

Let me compute its area. Similarly, 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 = -6 + 3 = -3 ⇒ (-6, -3)

Intersection of y=3x-9 and y=x+3: 3x -9 = x +3 ⇒ 2x = 12 ⇒ x = 6, y = 6 + 3 = 9 ⇒ (6, 9)

Intersection of y=3x-9 and y=x-9: 3x -9 = x -9 ⇒ 2x = 0 ⇒ x = 0, y = 0 -9 = -9 ⇒ (0, -9)

Vertices: (0,3), (-6, -3), (6,9), (0, -9). Let me order them properly. Maybe (0,3), (-6,-3), (0,-9), (6,9), back to (0,3). Let's apply shoelace.

Sum1 = 0*(-3) + (-6)*(-9) + 0*9 + 6*3 = 0 + 54 + 0 + 18 = 72

Sum2 = 3*(-6) + (-3)*0 + (-9)*6 + 9*0 = -18 + 0 -54 + 0 = -72

Area = 1/2 |72 - (-72)| = 1/2 * 144 = 72. Correct. So this also checks out. Therefore, the case with a=3, b=1, c=3, d=9 is valid and gives the total sum 3+1+3+9=16. But the problem says "the smallest possible value of a+b+c+d". However, maybe there is a configuration with smaller numbers?

Wait, let me think again. What if there are other solutions where (c + d)^2 / (d - c)^2 = 4, but with different relationships? For example, if c and d are not multiples of 3, but somehow fractions? But no, since c and d are positive integers.

Alternatively, maybe my formula for the area is incorrect? Let me check with another example. Suppose a=2, b=1, c=1, d=3. Then |a - b| = 1. Compute the area as |(d - c)^2 / (a - b)| = (3 -1)^2 / 1 = 4. Let me compute manually.

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

Vertices:

Intersection of 2x+1 and x+1: 2x +1 = x +1 ⇒ x=0, y=1 ⇒ (0,1)

Intersection of 2x+1 and x+3: 2x +1 = x +3 ⇒ x=2, y=5 ⇒ (2,5)

Intersection of 2x+3 and x+3: 2x +3 = x +3 ⇒ x=0, y=3 ⇒ (0,3)

Intersection of 2x+3 and x+1: 2x +3 = x +1 ⇒ x=-2, y=-2 +1 = -1 ⇒ (-2, -1)

Vertices: (0,1), (2,5), (0,3), (-2,-1). Shoelace formula:

Order: (0,1), (2,5), (0,3), (-2,-1), (0,1)

Sum1 = 0*5 + 2*3 + 0*(-1) + (-2)*1 = 0 + 6 + 0 -2 = 4

Sum2 = 1*2 + 5*0 + 3*(-2) + (-1)*0 = 2 + 0 -6 + 0 = -4

Area = 1/2 |4 - (-4)| = 4. Which matches the formula. Therefore, the formula is correct.

Therefore, in this case, with a=2, b=1, c=1, d=3, area is 4. If we use our previous equations:

For this case, (d - c)^2 / |a - b| = (2)^2 / 1 = 4, and (c + d)^2 / |a - b| = (4)^2 / 1 = 16. So if someone had 18 and 72, they would have to scale accordingly. But in our problem, we have the equations:

(d - c)^2 / k = 18

(c + d)^2 / k = 72

So if we have 18 and 72, then the ratio is 4, which is the same as the example. However, in this case, if we set (d - c)^2 / k = 18 and (c + d)^2 / k = 72, then (c + d)^2 / (d - c)^2 = 4, which as we saw, implies d = 3c or c = 3d. Therefore, in that case, k must be such that (d - c)^2 = 18k and (c + d)^2 = 72k. Which is the same as before. But in our previous example, k was 1. Therefore, if we want to have (d - c)^2 = 18k and (c + d)^2 = 72k, then (c + d)^2 = 4*(d - c)^2. Which is the same condition. Therefore, the same logic applies. Therefore, to find minimal a + b + c + d, we need to choose minimal c and d such that d = 3c or c = 3d, and k = (d - c)^2 / 18 is integer. Which for d = 3c, gives k = (2c)^2 / 18 = 4c² / 18 = 2c² / 9. Therefore, 2c² must be divisible by 9 ⇒ c must be divisible by 3. Let me let c = 3m ⇒ d = 9m. Then k = 2*(9m²)/9 = 2m². Therefore, |a - b| = 2m². To minimize a + b + c + d, we need minimal m, which is 1. Then, c = 3, d = 9, |a - b| = 2. Hence, the minimal a + b is 1 + 3 = 4 (if we take b = 1, a = 3). Therefore, total sum 4 + 3 + 9 = 16. Hence, 16 is the minimal.

But maybe there are other values where k is different? For example, suppose that (d - c)^2 = 18k and (c + d)^2 = 72k, but k is not necessarily equal to |a - b|. Wait, no, because we derived earlier that the area is |(d - c)^2 / (a - b)| = 18 and |(c + d)^2 / (a - b)| = 72, so k = |a - b|. Therefore, k must equal both (d - c)^2 / 18 and (c + d)^2 / 72. Therefore, these two expressions must be equal, which gives the ratio of 4. Hence, the same condition. Therefore, the minimal sum is 16.

But let me check with m = 1 and different a and b. For example, if |a - b| = 2, then possible a and b are (3,1), (4,2), (5,3), etc. For example, if we take a = 2, b = 0 (but b must be positive), so invalid. If a = 1, b = -1 (invalid). So only positive integers. Hence, the minimal a + b is 1 + 3 = 4. Hence, total sum 3 + 1 + 3 + 9 = 16.

But wait, maybe there is a case where the formula is different? For example, if the lines are not in the order I considered? For example, maybe the distance between the lines isn't |c - d| / sqrt(a² + 1), but something else?

Wait, but no, the distance between y = ax + c and y = ax + d is |c - d| / sqrt(a² + 1). Similarly for the other pair. Then the area of the parallelogram is the product of the distances divided by sin(theta), where theta is the angle between the two lines. Wait, but earlier I derived it as |(d - c)^2 / (a - b)|, but how is that related?

Alternatively, maybe I need to reconcile these two expressions. Let me think. The area can also be calculated as the distance between the first pair of lines multiplied by the distance between the second pair of lines divided by sin(theta), where theta is the angle between the two lines. Let me check that.

Distance between first pair: D1 = |c - d| / sqrt(a² + 1)

Distance between second pair: D2 = |c - d| / sqrt(b² + 1) ?

Wait no, first pair is y = ax + c and y = ax + d: distance is |c - d| / sqrt(a² + 1). The second pair is y = bx + c and y = bx + d: distance is |c - d| / sqrt(b² + 1). The angle theta between the two lines can be calculated as tan(theta) = |(m2 - m1)/(1 + m1m2)| = |(b - a)/(1 + ab)|. Then, the area of the parallelogram is D1 * D2 / sin(theta). Let me compute that:

Area = [ |c - d| / sqrt(a² + 1) ] * [ |c - d| / sqrt(b² + 1) ] / sin(theta). But this seems complicated. However, earlier we derived the area as |(d - c)^2 / (a - b)|. Let me check if these are equivalent.

Alternatively, since we have two expressions, let me consider our previous example with a=3, b=1, c=3, d=9. Let me compute using this formula.

D1 = |3 - 9| / sqrt(9 + 1) = 6 / sqrt(10)

D2 = |3 - 9| / sqrt(1 + 1) = 6 / sqrt(2)

sin(theta): tan(theta) = |(b - a)/(1 + ab)| = |(1 - 3)/(1 + 3*1)| = | -2 / 4 | = 1/2. Therefore, theta = arctan(1/2). So sin(theta) = 1 / sqrt(1 + 4) = 1 / sqrt(5). Therefore, Area = (6/sqrt(10)) * (6/sqrt(2)) / (1/sqrt(5)) ) = (36 / sqrt(20)) * sqrt(5) ) = 36 / (sqrt(20)/sqrt(5)) ) = 36 / sqrt(4) ) = 36 / 2 = 18. Which matches our previous result. Therefore, the formula using distances and sin(theta) gives the same result as the cross product formula.

Therefore, both formulas are equivalent. Therefore, the formula |(d - c)^2 / (a - b)| is correct.

Therefore, our previous reasoning is correct. Hence, the minimal possible value is 16. However, let me check again if there are other possibilities where k is different.

For example, if c and d are not multiples of 3, but the value of |a - b| is such that 2c² / 9 is integer. Let me suppose c is 3m, as before, but if c is not multiple of 3, say c = 6, then 2*(6)^2 /9 = 2*36/9 = 8, which is integer. But c=6 would give d=18, which would be larger. Hence, total sum would be higher. Therefore, m=1 gives minimal sum.

But wait, maybe there are other values for c and d not in the d=3c or c=3d relationship? For example, suppose that [(c + d)^2]/[(d - c)^2] = 4, but with different c and d. For instance, c=2, d=4. Then (c + d)/(d - c) = 6/2 = 3 ≠ 2. Doesn't satisfy. If c=4, d=6: (4 + 6)/(6 - 4) = 10/2 = 5 ≠ 2. Not. If c=1, d= 3: (1 + 3)/(3 - 1) = 4/2 = 2. Which satisfies. But then for this case, (d - c)^2 / k = 18 ⇒ 2² /k = 18 ⇒ k = 4 / 18 = 2/9, which is not integer. So invalid. Similarly, c=2, d= 6: (2 + 6)/(6 - 2) = 8/4 = 2. So this satisfies. Then check if k = (d - c)^2 / 18 = (4)^2 / 18 = 16 / 18 = 8/9 not integer. Not valid. c=4, d= 12: (4 + 12)/(12 - 4) = 16 / 8 = 2. Then (d - c)^2 / 18 = 8² / 18 = 64 / 18 = 32/9. Not integer. So only when c is multiple of 3, we can get integer k. Therefore, our previous conclusion holds.

Hence, the minimal possible value of a + b + c + d is 16. But wait, the problem says "positive integers", and in our case, all are positive. However, I want to check if there is a case where the sum is smaller. For example, let me think of other possibilities where the area formulas hold with different relations.

Suppose we have different values for k. For example, suppose that k divides both 18 and 72. But since 18 divides 72, k must divide 18. Wait, because (d - c)^2 / k = 18 and (c + d)^2 / k = 72 ⇒ k divides both 18 and 72. Therefore, k divides gcd(18, 72) = 18. Therefore, possible values of k are divisors of 18: 1, 2, 3, 6, 9, 18.

But from previous, k = (d - c)^2 / 18 must be equal to (c + d)^2 / 72. So let me denote:

Let me set k as a positive integer that divides 18, so possible k = 1, 2, 3, 6, 9, 18. Let me check for each possible k:

For each k, we have:

From first equation: (d - c)^2 = 18k

From second equation: (c + d)^2 = 72k

Therefore, (c + d)^2 = 4*(d - c)^2 ⇒ which gives us the same relation as before. So for each k, we need to solve (c + d)^2 = 4*(d - c)^2 ⇒ same as before, which gives d = 3c or c = 3d. Therefore, for each k, solving:

Case 1: d = 3c

Then, (d - c)^2 = (2c)^2 = 4c² = 18k ⇒ c² = (18k)/4 = (9k)/2

Similarly, since c must be integer, 9k must be divisible by 2 ⇒ k must be even. Since k divides 18, k can be 2, 6, 18.

If k = 2: c² = 9*2 / 2 = 9 ⇒ c = 3 ⇒ d = 9. Which is our previous case, leading to a + b + c + d = 16.

If k = 6: c² = 9*6 / 2 = 27 ⇒ c² = 27 ⇒ c is not integer.

If k = 18: c² = 9*18 / 2 = 81 ⇒ c = 9 ⇒ d = 27. Then, |a - b| = k = 18. Therefore, a and b are such that |a - b| = 18. To minimize a + b, take b = 1, a = 19. Then total sum is 19 + 1 + 9 + 27 = 56, which is larger than 16. Hence, this is worse.

Case 2: c = 3d

Then, (d - c)^2 = (d - 3d)^2 = (-2d)^2 = 4d² = 18k ⇒ d² = (18k)/4 = (9k)/2. Similarly, need 9k divisible by 2 ⇒ k even. So k = 2, 6, 18.

For k = 2: d² = 9*2 / 2 = 9 ⇒ d = 3 ⇒ c = 9. Which is same as previous case, leading to total sum 16.

For k = 6: d² = 9*6 / 2 = 27 ⇒ d not integer.

For k = 18: d² = 9*18 / 2 = 81 ⇒ d = 9 ⇒ c = 27. Then |a - b| = 18, leading to a + b + c + d = same as above, 56.

Hence, only when k = 2, we get valid integers c and d, leading to total sum 16. Therefore, 16 is indeed minimal.

Wait, but let me check if there are other k values if we don't restrict ourselves to d = 3c or c = 3d. However, we know that the equation (c + d)^2 = 4*(d - c)^2 only has those two solutions. Therefore, no other possibilities.

Therefore, the minimal possible value of a + b + c + d is 16. However, let me check if the problem specifies that a, b, c, d are positive integers, and in our case they are. However, the problem says "smallest possible value of a + b + c + d". Therefore, 16 is the answer?

But let me check if there is another possible k. For example, if k is not a divisor of 18? Wait, earlier I thought that since k divides 18 from (d - c)^2 = 18k, but actually, no. Wait, k is |a - b|, which is a positive integer. The equations are:

(d - c)^2 = 18k and (c + d)^2 = 72k. Therefore, since (c + d)^2 = 4*(d - c)^2, which is 4*18k = 72k, so it's consistent. Therefore, k can be any positive integer such that (d - c)^2 = 18k has integer solutions for d and c, with c and d positive integers, and similarly for (c + d)^2 = 72k. However, the key point is that for (d - c)^2 = 18k and (c + d)^2 = 72k, we must have that 18k is a perfect square, and 72k is also a perfect square.

Wait, hold on! This might be another way to look at it. Since (d - c)^2 = 18k ⇒ 18k must be a perfect square. Similarly, 72k must be a perfect square. Let me denote:

Let me let 18k = m² and 72k = n². Then, n² = 72k = 4*18k = 4m² ⇒ n = 2m. Therefore, the relationship between m and n is maintained. Therefore, 18k must be a perfect square. Therefore, prime factorization of 18k must have even exponents.

Since 18 = 2 * 3². Therefore, for 18k to be a perfect square, k must supply the missing factors to make all exponents even. Therefore, k must be of the form 2 * t², where t is a positive integer. Because:

18k = 2 * 3² * k. For this to be square, the exponent of 2 must be even, so k must have an odd exponent of 2. Therefore, k must contain a factor of 2, and other primes must be squares. Let me write k = 2 * t². Therefore, 18k = 18*2*t² = 36t² = (6t)², which is a perfect square. Then, 72k = 72*2*t² = 144t² = (12t)², which is also a perfect square. Therefore, k must be twice a square. Therefore, k = 2t², where t is a positive integer. Therefore, substituting back into (d - c)^2 = 18k = 18*2t² = 36t² ⇒ d - c = 6t. Similarly, (c + d)^2 = 72k = 72*2t² = 144t² ⇒ c + d = 12t. Therefore, we have:

c + d = 12t

d - c = 6t

Solving these equations:

Adding both equations:

2d = 18t ⇒ d = 9t

Subtracting the second equation from the first:

2c = 6t ⇒ c = 3t

Therefore, c = 3t and d = 9t. Therefore, this is the same result as before. Hence, this is the general solution. Therefore, c = 3t, d = 9t. Then, |a - b| = k = 2t².

Therefore, to minimize a + b + c + d, we need to choose the smallest possible t. Since t is a positive integer, the minimal t is 1. Therefore, c = 3, d = 9, and |a - b| = 2*1² = 2. To minimize a + b, choose a and b such that their difference is 2 and they are positive integers. The minimal sum occurs when b = 1 and a = 3, so a + b = 4. Hence, total sum is a + b + c + d = 3 + 1 + 3 + 9 = 16.

Therefore, this confirms our previous conclusion. Therefore, the smallest possible value is 16. However, the problem says "the smallest possible value of a + b + c + d". But I need to check if there are any constraints that I missed. For example, are there any restrictions on a and b? The problem states that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9, all positive integers. So this is valid.

But just to be thorough, let me check if there are other values of t which might allow smaller sums. For example, if t = 1/2, but t must be integer. Hence, no. Therefore, t = 1 is the minimum. Therefore, 16 is the minimal sum.

But let me check the problem statement again. It says the parallelogram bounded by the lines y=ax+c, y=ax+d, y=bx+c, and y=bx+d has area 18. Similarly for the other one. Is there any possibility that the values could be different if the lines are arranged differently? For example, maybe the lines could intersect in a different way? But no, because given the slopes and intercepts, the intersections are uniquely determined.

Therefore, I think 16 is correct. However, I feel uncertain because sometimes problems have trick answers or I might have missed something. Let me check if there's a configuration with smaller numbers. For example, take t = 1, sum 16. Suppose someone tried t = 1, but with different a and b. For example, if a - b = 2, and instead of a = 3, b = 1, maybe a = 2, b = 0. But b must be positive, so b = 0 is invalid. Or a = 4, b = 2. Then a + b = 6, and total sum 6 + 3 + 9 = 18, which is larger than 16. Similarly, if a = 1, b = -1, but b must be positive. So no. Therefore, no smaller sum.

Therefore, I think the answer is 16. However, I recall that in some cases, problems might have different configurations where the area formula differs, but according to our derivation and verification with examples, this seems correct.

Wait, but to be absolutely sure, let me check if there's a way to have smaller numbers by considering that the formula for area is |(d - c)^2 / (a - b)|, and maybe |a - b| is larger but allows smaller c and d. For example, suppose |a - b| is larger, but c and d are smaller such that (d - c)^2 / |a - b| = 18 and (c + d)^2 / |a - b| = 72. For instance, suppose |a - b| = 9, then (d - c)^2 = 18*9 = 162, which is not a perfect square. Similarly, |a - b| = 8: (d - c)^2 = 144 ⇒ d - c = 12, and (c + d)^2 = 72*8 = 576 ⇒ c + d = 24. Then solving d - c = 12 and c + d = 24 ⇒ d = 18, c = 6. Then, check if this is valid. So c = 6, d = 18, |a - b| = 8. Then, a + b + c + d = a + b + 6 + 18 = a + b + 24. To minimize a + b, with |a - b| = 8. Minimal sum of a + b occurs when a = 1 and b = 9 (since 1 + 9 = 10), or a = 9 and b = 1. Hence, total sum 10 + 24 = 34, which is larger than 16. Therefore, worse.

Alternatively, take |a - b| = 18. Then (d - c)^2 = 18*18 = 324 ⇒ d - c = 18, and (c + d)^2 = 72*18 = 1296 ⇒ c + d = 36. Solving: d = c + 18, so 2c + 18 = 36 ⇒ 2c = 18 ⇒ c = 9, d = 27. Then, a + b + c + d = a + b + 36. Minimal a + b is 1 + 19 = 20, total sum 56. Larger than 16.

Alternatively, take |a - b| = 3: (d - c)^2 = 18*3 = 54, not square. Similarly for |a - b| = 4: (d - c)^2 = 72, which is not square. |a - b| = 6: (d - c)^2 = 108, not square. |a - b| = 12: (d - c)^2 = 216, not square. Hence, no improvement.

Therefore, indeed, the minimal sum is 16.

But just to check once more, if there's another case where c + d and d - c are multiples that give smaller sums. For example, suppose we take t = 1, which gives c = 3, d = 9. If there's a smaller t? t must be positive integer, so t=1 is the smallest. Hence, no.

Therefore, I believe the correct answer is 16. But to make sure, let me check once more with different a and b. Suppose I take a = 1, b = -1. But b must be positive. Not allowed. a = 4, b = 2: sum a + b = 6. c = 3, d = 9. Total sum 6 + 3 + 9 = 18. Larger than 16. So no.

Yes, 16 is the minimal.

**Final Answer**
\boxed{16}

To solve the problem, we consider the two parallelograms defined by certain lines. Both parallelograms are constructed from lines with slopes $ a $ and $ b $, and intercepts involving $ c $ and $ d $. These lines generate parallelograms whose areas are given as 18 and 72 respectively. Our goal is to minimize the sum $ a + b + c + d $, where all variables are positive integers.

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### Key Observations:

1. **Area of a Parallelogram Defined by Parallel Lines:**

   The area of a parallelogram formed by the lines
   $ y = ax + c $, $ y = ax + d $, $ y = bx + c $, and $ y = bx + d $
   can be computed using:
   $$
   \text{Area} = \frac{(d - c)^2}{|a - b|}
   $$

   Similarly, the second parallelogram (with lines $ y = ax + c $, $ y = ax - d $, $ y = bx + c $, and $ y = bx - d $) gives an area of:
   $$
   \text{Area} = \frac{(c + d)^2}{|a - b|}
   $$

2. **Setting Up Equations:**
   From the problem, we are given:
   $$
   \frac{(d - c)^2}{|a - b|} = 18 \quad \text{and} \quad \frac{(c + d)^2}{|a - b|} = 72
   $$
   Dividing these equations gives:
   $$
   \frac{(c + d)^2}{(d - c)^2} = 4 \Rightarrow \frac{c + d}{d - c} = 2
   $$
   Solving this ratio yields two cases:
   - $ d = 3c $
   - $ c = 3d $

3. **Choosing the Case:**
   We'll choose the case $ d = 3c $ (the symmetric case $ c = 3d $ will be identical in terms of minimization). Substituting $ d = 3c $ into the first area formula:
   $$
   \frac{(3c - c)^2}{|a - b|} = \frac{(2c)^2}{|a - b|} = \frac{4c^2}{|a - b|} = 18
   \Rightarrow |a - b| = \frac{4c^2}{18} = \frac{2c^2}{9}
   $$
   For this to be an integer, $ 2c^2 $ must be divisible by 9. The smallest such $ c $ that satisfies this is $ c = 3 $, so $ c = 3 $, $ d = 9 $, and:
   $$
   |a - b| = \frac{2 \cdot 3^2}{9} = \frac{18}{9} = 2
   $$

4. **Minimizing $ a + b + c + d $:**
   To minimize the total sum, we choose the smallest possible positive integers $ a $ and $ b $ such that $ |a - b| = 2 $. The pair $ a = 3 $, $ b = 1 $ gives the minimal sum $ a + b = 4 $.

   So the total sum is:
   $$
   a + b + c + d = 3 + 1 + 3 + 9 = \boxed{16}
   $$

---

### Final Answer:

$$
\boxed{16}
$$

[Assistant]
Explanation:
Okay, so I need to find the smallest possible value of a + b + c + d where a, b, c, d are positive integers. The problem gives two parallelograms with areas 18 and 72, defined by different sets of lines. Let me try to visualize and understand the problem step by step.

First, let me recall that the area of a parallelogram can be calculated if we know the base and the height, but since these parallelograms are defined by lines, maybe there's a formula related to the distance between lines and the angle between them? Hmm.

The first parallelogram is bounded by y = ax + c, y = ax + d, y = bx + c, and y = bx + d. Similarly, the second one is bounded by y = ax + c, y = ax - d, y = bx + c, and y = bx - d. The areas are given as 18 and 72 respectively.

Since these are parallelograms formed by two pairs of parallel lines, each pair of lines with the same slope (a and b). The distance between the lines y = ax + c and y = ax + d should be the same for both pairs, but since the second parallelogram has different constants (c and -d), maybe the distances are related?

Wait, let me think. For two parallel lines y = mx + c1 and y = mx + c2, the distance between them is |c1 - c2| / sqrt(m² + 1). So for the first parallelogram, the vertical distances between the lines would be |c - d| / sqrt(a² + 1) and |c - d| / sqrt(b² + 1)? Wait, no, actually, since both pairs of lines are parallel. The distance between 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). But since it's a parallelogram, the area would be the product of these two distances multiplied by the sine of the angle between the lines? Wait, no, actually, if we have two pairs of lines, the area of the parallelogram is the product of the distances between the two pairs times the sine of the angle between the two lines? Wait, maybe?

Alternatively, maybe the area can be calculated by the distance between one pair of lines multiplied by the distance between the other pair divided by sin(theta), where theta is the angle between the lines? Wait, I need to recall the formula for area of a parallelogram formed by two pairs of lines.

Alternatively, think of the vectors. If we have two sets of lines, the sides of the parallelogram can be determined by the direction vectors of the lines. Let me try this approach.

Let me take the first parallelogram: lines y = ax + c, y = ax + d (both with slope a), and lines y = bx + c, y = bx + d (both with slope b). The intersection points of these lines will form the vertices of the parallelogram. Let me find the coordinates of these vertices.

First, find intersection of y = ax + c and y = bx + c. Setting ax + c = bx + c => ax = bx => (a - b)x = 0. Since a ≠ b (otherwise, they would be parallel lines and not form a parallelogram), so x = 0. Then y = a*0 + c = c. So one vertex is (0, c).

Second, intersection of y = ax + c and y = bx + d. Set ax + c = bx + d => (a - b)x = d - c => x = (d - c)/(a - b). Then y = a*( (d - c)/(a - b) ) + c = [a(d - c) + c(a - b)] / (a - b) = [ad - ac + ac - bc]/(a - b) = (ad - bc)/(a - b). Let me write that as y-coordinate.

Third, intersection of y = ax + d and y = bx + c. Similarly, ax + d = bx + c => (a - b)x = c - d => x = (c - d)/(a - b). Then y-coordinate: same as above, but with sign switched? Let's compute:

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).

Fourth, intersection of y = ax + d and y = bx + d. Setting ax + d = bx + d => (a - b)x = 0 => x = 0, then y = d. So fourth vertex is (0, d).

Wait, so the four vertices are (0, c), [(d - c)/(a - b), (ad - bc)/(a - b)], [(c - d)/(a - b), (ac - bd)/(a - b)], and (0, d). Wait, but [(c - d)/(a - b), (ac - bd)/(a - b)] is equivalent to [-(d - c)/(a - b), (ac - bd)/(a - b)]. Let me check if this makes sense.

Alternatively, maybe I can think of vectors between these points to compute the area. But perhaps there's a formula for the area of a parallelogram formed by two pairs of lines. Let me recall that.

In general, if you have two pairs of lines: one pair with slope m1: y = m1x + c1 and y = m1x + c2, and another pair with slope m2: y = m2x + d1 and y = m2x + d2. Then the area of the parallelogram formed by these lines is |(c2 - c1)(d2 - d1)| / |m1 - m2|. Wait, no, that seems too simple. Wait, maybe?

Wait, actually, the distance between the first pair of lines is |c1 - c2| / sqrt(m1² + 1), and the distance between the second pair is |d1 - d2| / sqrt(m2² + 1). But the area is the product of these distances divided by the sine of the angle between the lines? Wait, no. Wait, in general, if you have two vectors, the area is |v × w|. So maybe we need to find vectors along the sides of the parallelogram and take the cross product.

Let me try that.

For the first parallelogram, with lines y = ax + c, y = ax + d, y = bx + c, y = bx + d. Let me find two vectors that form the sides of the parallelogram.

Take the vertex (0, c) and move along the direction of the line y = ax + c, which has direction vector (1, a). Similarly, moving along y = bx + c, direction vector (1, b). Wait, but actually, the sides of the parallelogram are the directions of the lines? Wait, maybe not. Let me think.

Alternatively, the vectors representing the sides of the parallelogram can be found by taking the difference between two adjacent vertices. For example, from (0, c) to the intersection point [(d - c)/(a - b), (ad - bc)/(a - b)], which I can call point A. Let me compute vector OA - vector OB.

Wait, maybe it's easier to compute the vectors. Let me take two adjacent sides of the parallelogram. For example, from (0, c) to [(d - c)/(a - b), (ad - bc)/(a - b)] and from (0, c) to [(c - d)/(a - b), (ac - bd)/(a - b)]. Wait, but maybe I need to find two vectors that are along the sides. Alternatively, since the sides are between the lines, perhaps the vectors can be determined by the direction vectors of the lines?

Alternatively, maybe we can use the formula for the area of a parallelogram defined by two lines. Wait, I found a resource once that mentioned that if you have two sets of parallel lines, the area can be calculated as (distance between first pair) * (distance between second pair) / sin(theta), where theta is the angle between the two sets of lines. Wait, but why divided by sin(theta)?

Wait, suppose you have two vectors forming adjacent sides of a parallelogram. The area is the magnitude of their cross product. If the two lines are at an angle theta, then the cross product would involve sin(theta). Let me think.

Alternatively, let me think of the area as the base times height. Suppose one pair of lines defines the base, and the other pair defines the height. Wait, but the height would depend on the angle between the lines.

But maybe there's a formula here. Let me try to recall. Suppose we have two lines with slopes m1 and m2. The angle theta between them is given by tan(theta) = |(m2 - m1)/(1 + m1m2)|. Then, if I can find the distances between the lines and multiply them by sin(theta), that would give the area?

Wait, if we take the distance between the first pair of lines, which is D1, and the distance between the second pair, D2, then the area of the parallelogram should be D1 * D2 / sin(theta)? Wait, no. Wait, if theta is the angle between the lines, then if you have a base of length D1 and a height corresponding to the other side, which is D2 / sin(theta)?

This is getting confusing. Maybe I need to use coordinates.

Alternatively, since I have the coordinates of the vertices, maybe I can use the shoelace formula to compute the area. Let me try that.

First, for the first parallelogram. Let me denote the four vertices:

Vertex 1: (0, c)

Vertex 2: [(d - c)/(a - b), (ad - bc)/(a - b)]

Vertex 3: [(c - d)/(a - b), (ac - bd)/(a - b)]

Vertex 4: (0, d)

Wait, but if I connect these points, is that a parallelogram? Let me check if opposite sides are equal and parallel. Let me compute the vectors between these points.

Vector from Vertex 1 to Vertex 2: [(d - c)/(a - b) - 0, (ad - bc)/(a - b) - c] = [(d - c)/(a - b), (ad - bc - c(a - b))/(a - b)] = [(d - c)/(a - b), (ad - bc - ac + bc)/(a - b)] = [(d - c)/(a - b), (ad - ac)/(a - b)] = [(d - c)/(a - b), a(d - c)/(a - b)] = (d - c)/(a - b) * (1, a)

Similarly, vector from Vertex 4 to Vertex 3: [(c - d)/(a - b) - 0, (ac - bd)/(a - b) - d] = [(c - d)/(a - b), (ac - bd - d(a - b))/(a - b)] = [(c - d)/(a - b), (ac - bd - ad + bd)/(a - b)] = [(c - d)/(a - b), (ac - ad)/(a - b)] = [(c - d)/(a - b), a(c - d)/(a - b)] = (c - d)/(a - b)*(1, a) = -(d - c)/(a - b)*(1, a)

Which is exactly the negative of the vector from Vertex 1 to Vertex 2, so that makes sense for a parallelogram. Similarly, vectors between other sides.

Similarly, vector from Vertex 1 to Vertex 4 is (0 - 0, d - c) = (0, d - c). Then, vector from Vertex 2 to Vertex 3: [(c - d)/(a - b) - (d - c)/(a - b), (ac - bd)/(a - b) - (ad - bc)/(a - b)] = [ (c - d - d + c)/(a - b), (ac - bd - ad + bc)/(a - b) ] = [ (2c - 2d)/(a - b), (ac - ad + bc - bd)/(a - b) ] = [ 2(c - d)/(a - b), a(c - d) + b(c - d) / (a - b) ] = [ 2(c - d)/(a - b), (a + b)(c - d)/(a - b) ] = (c - d)/(a - b)*(2, a + b)

Wait, but the vector from Vertex 1 to Vertex 4 is (0, d - c). Let me check if this is parallel to the vector from Vertex 2 to Vertex 3. The vector from Vertex 1 to Vertex 4 is (0, d - c), and the vector from Vertex 2 to Vertex 3 is (c - d)/(a - b)*(2, a + b). For them to be parallel, there must be a scalar multiple. Let me see:

(0, d - c) = k*(2(c - d)/(a - b), (a + b)(c - d)/(a - b))

But (d - c) = -(c - d), so:

(0, - (c - d)) = k*(2(c - d)/(a - b), (a + b)(c - d)/(a - b))

So, if we let k = - (a - b)/2, then:

Left side: (0, - (c - d))

Right side: - (a - b)/2 * [ 2(c - d)/(a - b), (a + b)(c - d)/(a - b) ) ] = - (a - b)/2 * 2(c - d)/(a - b) , - (a - b)/2 * (a + b)(c - d)/(a - b) ) = - (c - d), - (a + b)(c - d)/2 )

Wait, that doesn't match the left side. So maybe I made a mistake here. Wait, the vector from Vertex 1 to Vertex 4 is (0, d - c). The vector from Vertex 2 to Vertex 3 is something else. Maybe I need to find another vector.

Alternatively, maybe I can use the shoelace formula. Let me list the coordinates of the vertices in order. Let me suppose they are ordered as follows: (0, c), [(d - c)/(a - b), (ad - bc)/(a - b)], (0, d), [(c - d)/(a - b), (ac - bd)/(a - b)]? Wait, not sure of the order. Maybe it's better to go around the parallelogram.

Alternatively, maybe I can compute the area using vectors. The area is |(v1 × v2)|, where v1 and v2 are vectors representing adjacent sides.

Earlier, I found that one side vector is (d - c)/(a - b)*(1, a), and another side vector can be found from (0, d - c). Wait, but the vector from Vertex 1 (0, c) to Vertex 4 (0, d) is (0, d - c). So if one vector is (d - c)/(a - b)*(1, a) and another vector is (0, d - c), then the cross product (in 2D, the magnitude is |v1_x v2_y - v1_y v2_x|).

Let me compute that. Let me denote vector u = [(d - c)/(a - b), a(d - c)/(a - b)] and vector v = [0, d - c]

Then cross product magnitude |u_x v_y - u_y v_x| = | [(d - c)/(a - b)*(d - c) - a(d - c)/(a - b)*0 ] | = | (d - c)^2 / (a - b) |. Therefore, the area of the parallelogram is |(d - c)^2 / (a - b)|. Wait, but the area is given as 18.

Wait, so according to this, Area1 = |(d - c)^2 / (a - b)| = 18. Similarly, for the second parallelogram, which is bounded by y = ax + c, y = ax - d, y = bx + c, y = bx - d. Let me compute its area similarly.

So for the second parallelogram, the constants are c and -d instead of c and d. Following the same logic, the distance between the lines y = ax + c and y = ax - d is |c - (-d)| / sqrt(a² + 1) = |c + d| / sqrt(a² + 1). But wait, but according to the previous calculation, the area is |(difference in constants)^2 / (a - b)|. Wait, in the first case, the difference between the constants was (d - c), but here, the difference would be (-d - c) = -(c + d). Then, the area would be |(- (c + d))² / (a - b)| = |(c + d)^2 / (a - b)| = 72.

Wait, but according to the previous calculation, for the first parallelogram, the area was |(d - c)^2 / (a - b)| = 18, and for the second one, it's |(c + d)^2 / (a - b)| = 72. Therefore, we have two equations:

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

2. (c + d)^2 / |a - b| = 72

Since all variables are positive integers, and a and b are positive integers, but we don't know if a > b or b > a. However, since the denominators are absolute values, we can assume that |a - b| is positive, so the denominators are same for both equations. Therefore, we can denote k = |a - b|, so that:

(d - c)^2 / k = 18

(c + d)^2 / k = 72

Therefore, if we take the ratio of the second equation to the first, we get [(c + d)^2 / k] / [(d - c)^2 / k] = 72 / 18 = 4. Therefore, [(c + d)^2] / [(d - c)^2] = 4 => (c + d)/(d - c) = 2 (since c + d and d - c are positive? Wait, since c and d are positive integers, but we need to check if d - c is positive. If d - c is positive, then (c + d)/(d - c) = 2 => c + d = 2(d - c) => c + d = 2d - 2c => c + d - 2d + 2c = 0 => 3c - d = 0 => d = 3c. Alternatively, if d - c is negative, then |d - c| is positive. However, since (d - c)^2 is positive, and the equations are in terms of absolute value, but when we took the ratio, we assumed (d - c)^2 is positive, so regardless, the ratio gives [(c + d)^2]/[(d - c)^2] = 4. Therefore, taking square roots, (c + d)/|d - c| = 2. Since c and d are positive integers, let's suppose d > c (so that d - c is positive). Therefore, (c + d)/(d - c) = 2 => c + d = 2d - 2c => 3c = d.

Alternatively, if d < c, then |d - c| = c - d, so (c + d)/(c - d) = 2 => c + d = 2c - 2d => 3d = c. But since c and d are positive integers, both cases are possible. However, since the problem states that a, b, c, d are positive integers, and we need to find the minimal sum, we need to consider both cases.

So, first case: d = 3c. Second case: c = 3d. Let me check both possibilities.

Let me consider case 1: d = 3c.

Substituting back into the first equation:

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

Similarly, from the second equation:

(c + d)^2 / k = 72 => (c + 3c)^2 / k = 72 => (4c)^2 / k = 72 => 16c² / k = 72 => 2c² / k = 9. Same as the first equation. So consistent.

Therefore, k = 2c² / 9. But k is |a - b|, which must be a positive integer. Therefore, 2c² must be divisible by 9, so c² must be divisible by 9/ gcd(2, 9) = 9, since 2 and 9 are coprime. Therefore, c² must be divisible by 9 => c must be divisible by 3. Let me write c = 3m, where m is a positive integer. Then, d = 3c = 9m. Then, k = 2*(3m)^2 / 9 = 2*9m² / 9 = 2m². Therefore, |a - b| = 2m². Since a and b are positive integers, we can let a = b + 2m² or b = a + 2m². But since a and b are positive integers, both possibilities are allowed. However, to minimize a + b + c + d, we need to choose m as small as possible.

Similarly, in case 2: c = 3d. Then, substituting into first equation:

(d - c)^2 / k = 18 => (d - 3d)^2 / k = (-2d)^2 / k = 4d² / k = 18 => 2d² / k = 9 => k = 2d² / 9. Similarly, second equation:

(c + d)^2 / k = (3d + d)^2 / k = (4d)^2 / k = 16d² / k = 72 => 2d² / k = 9, same as before. Therefore, k = 2d² / 9. Then, since c = 3d, and c must be divisible by 3, which it is. Similarly, d must be such that 2d² is divisible by 9. Therefore, d must be divisible by 3. Let me set d = 3n, then c = 3d = 9n. Then k = 2*(3n)^2 / 9 = 2*9n² /9 = 2n². Therefore, |a - b| = 2n². Again, same as case 1.

Therefore, in both cases, the structure is similar: either c = 3m and d = 9m, or c = 9n and d = 3n, with |a - b| = 2m² or 2n². Therefore, to minimize a + b + c + d, we need to minimize m (or n) and choose between cases.

But note that in case 1, c = 3m, d = 9m, and in case 2, c = 9n, d = 3n. Let me compare. For case 1, the sum c + d = 3m + 9m = 12m. For case 2, c + d = 9n + 3n = 12n. So they are the same for same m or n. Therefore, which case is better? Well, since we need to minimize the total sum a + b + c + d. Let's consider case 1: c = 3m, d = 9m. Then, |a - b| = 2m². So to minimize a + b, given that |a - b| = 2m². Let me let a and b be such that a - b = 2m² or b - a = 2m². To minimize a + b, we can set one variable as small as possible. For example, let me assume a > b, so a = b + 2m². Then, a + b = 2b + 2m². To minimize this, set b as small as possible. Since b is a positive integer, minimal b is 1. Therefore, a = 1 + 2m², and then a + b = 2 + 2m².

Similarly, in case 2, c = 9n, d = 3n. Then |a - b| = 2n². Similarly, minimal a + b would be 2 + 2n². Therefore, in both cases, the sum a + b is 2 + 2m² or 2 + 2n². However, c + d is 12m in case 1 or 12n in case 2. Therefore, total sum is (2 + 2m²) + 12m or (2 + 2n²) + 12n. So, which gives a smaller total? Let's see for minimal m and n.

Since m and n are positive integers, starting with m = 1, n = 1.

Case 1, m = 1: total sum = 2 + 2(1)^2 + 12*1 = 2 + 2 + 12 = 16

Case 2, n = 1: total sum = 2 + 2(1)^2 + 12*1 = same 16

But we need to check if this is possible. Let me check if a, b, c, d are positive integers.

For case 1: m = 1, c = 3*1 = 3, d = 9*1 = 9. Then |a - b| = 2*(1)^2 = 2. Let me choose b = 1, then a = b + 2 = 3. Therefore, a = 3, b = 1. All positive integers. Then check if the given areas hold. Let's verify.

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. Therefore, this works. So total sum is a + b + c + d = 3 + 1 + 3 + 9 = 16. Wait, 3 + 1 is 4, 3 + 9 is 12, total 16. However, let me check case 2 with n = 1. c = 9*1 = 9, d = 3*1 = 3. Then |a - b| = 2*(1)^2 = 2. Let me choose b = 1, then a = 1 + 2 = 3. Then sum a + b + c + d = 3 + 1 + 9 + 3 = same 16. So same result.

But wait, the problem says that a, b, c, d are positive integers. However, in case 1, we have a = 3, b = 1, c = 3, d = 9. But the lines in the first parallelogram are y = ax + c = 3x + 3, y = 3x + 9, y = bx + c = 1x + 3, y = 1x + 9. The second parallelogram is y = 3x + 3, y = 3x - 9, y = x + 3, y = x - 9. Wait, but the second parallelogram is bounded by y = ax + c, y = ax - d, y = bx + c, y = bx - d. Here, ax - d would be 3x - 9, which is valid. Similarly, bx - d is x - 9, which is okay. So, that seems okay. So, total sum is 16. However, the question is asking for the smallest possible value of a + b + c + d. But 16 seems low. Wait, but maybe there's a way to get a smaller sum?

Wait, maybe m or n can be 0? But m and n are positive integers, so starting at 1. But maybe there's another case?

Wait, but in the ratio we had (c + d)/(d - c) = 2 or similar, leading to d = 3c or c = 3d. But is there another possibility? For example, if d - c is negative, then |d - c| = c - d. But when taking the ratio, we had [(c + d)^2]/[(d - c)^2] = 4, so [(c + d)/|d - c|]^2 = 4. Therefore, (c + d)/|d - c| = 2 or -2. But since c and d are positive integers, c + d is positive, and |d - c| is positive. Therefore, (c + d)/|d - c| = 2. Therefore, only that case.

So the only possibilities are d = 3c or c = 3d. So we need to check if there are other solutions. But in both cases, when m = 1, we get the total sum as 16. However, maybe there's a case where |a - b| is smaller? Let me check.

Wait, for example, if in case 1: d = 3c, and k = |a - b| = 2c² / 9. Since k must be integer, 2c² must be divisible by 9. Therefore, c must be divisible by 3, which we already considered. So if c = 3m, then k = 2*(3m)^2 / 9 = 2*9m² / 9 = 2m². Therefore, k = 2m². Similarly for case 2. Therefore, with m = 1, k = 2, which is integer. So that's okay.

But is there another way? Suppose that c and d are such that (c + d)^2 / (d - c)^2 = 4, but with different relationships between c and d. For example, if d - c = something else? Wait, but algebraically we have that [(c + d)/(d - c)] = 2, which gives d = 3c or c = 3d. So no other possibilities. Hence, the minimal m or n is 1, leading to total sum 16. However, the answer might be different. Wait, wait, but maybe there is another approach where the area is calculated differently?

Wait, earlier I derived the area as |(d - c)^2 / (a - b)|. Wait, but I need to check that again. Let me go back. When I calculated the cross product earlier, I had vectors u = [(d - c)/(a - b), a(d - c)/(a - b)] and v = [0, d - c]. Then cross product was |u_x v_y - u_y v_x| = | [(d - c)/(a - b)*(d - c) - a(d - c)/(a - b)*0 ]| = | (d - c)^2 / (a - b) |. Therefore, the area is |(d - c)^2 / (a - b)|. But is this correct? Let me check with a simple example.

Suppose a = 1, b = 0, c = 1, d = 2. Then lines y = x + 1, y = x + 2, y = 0x + 1, y = 0x + 2. The parallelogram is between these lines. The area should be |(2 - 1)^2 / (1 - 0)| = 1/1 = 1. Let me compute it manually. The vertices are (0, 1), intersection of y = x + 1 and y = 0x + 2: x + 1 = 2 => x = 1, y = 2. So (1, 2). Similarly, intersection of y = x + 2 and y = 0x + 1: x + 2 = 1 => x = -1, y = 1. So (-1, 1). And intersection of y = x + 1 and y = 0x + 1 is (0, 1). Wait, so vertices are (0, 1), (1, 2), (0, 2), (-1, 1). Using shoelace formula:

Coordinates: (0,1), (1,2), (0,2), (-1,1), back to (0,1).

Compute area: 1/2 |(0*2 + 1*2 + 0*1 + (-1)*1) - (1*1 + 2*0 + 2*(-1) + 1*0)|

Wait, let me use the shoelace formula step by step:

List the coordinates:

(0,1), (1,2), (0,2), (-1,1), (0,1)

Compute sum of x_i y_{i+1}:

0*2 + 1*2 + 0*1 + (-1)*1 = 0 + 2 + 0 -1 = 1

Compute sum of y_i x_{i+1}:

1*1 + 2*0 + 2*(-1) + 1*0 = 1 + 0 - 2 + 0 = -1

Subtract: 1 - (-1) = 2. Take absolute value and divide by 2: |2| / 2 = 1. Which matches the formula result. Therefore, my formula seems correct. Therefore, for the first case, area is |(d - c)^2 / (a - b)|. Therefore, previous calculations are correct.

Therefore, in our first case, the minimal total sum is 16. However, maybe there's a mistake here? Let me check if the problem says that the lines are "bounded by" the given lines. Is there a possibility that when we take a and b, they might need to satisfy some other condition? For example, if a = b, then the lines would be parallel, but since the problem says they form a parallelogram, a ≠ b must hold. Which they do in our case, since |a - b| = 2m² ≠ 0.

But let me check if a and b are positive integers. In our case, for m = 1, a = 3, b = 1, which are positive integers. Similarly, c = 3, d = 9. So all are positive integers. Therefore, 16 is possible. However, the problem says "the smallest possible value of a + b + c + d". Is 16 indeed the minimal? Or maybe there is a smaller sum with different relationships?

Wait, but maybe there is a case where (c + d)^2 / (d - c)^2 = 4 is satisfied with different c and d? For example, if c + d = 2(d - c), which gives d = 3c, or c + d = -2(d - c), but since all are positive, the second equation would lead to c + d = -2d + 2c => 3d = c. So same as before.

Alternatively, could there be cases where the ratio is 4 but with different relations? For instance, if (c + d)^2 = 4(d - c)^2. Then sqrt of both sides: c + d = ±2(d - c). Since all are positive, c + d = 2(d - c) => d = 3c or c + d = -2(d - c) => c + d = -2d + 2c => 3d = c. So again same results. Therefore, no other possibilities. Therefore, minimal is 16? But let me check.

Wait, but let me check with m = 1 gives sum 16. However, let me check if there is a case with m = 1, but different a and b. For example, if a and b are such that |a - b| = 2, but different from a = 3, b = 1. For example, a = 2, b = 0. But b must be positive integer, so b = 0 is invalid. Similarly, a = 4, b = 2. Then a + b = 6. But c = 3, d = 9, so total sum would be 4 + 2 + 3 + 9 = 18, which is larger than 16. Hence, the minimal in case 1 is when a = 3, b = 1, giving sum 16.

But maybe there are other possibilities where the ratio is 4, but with different values?

Alternatively, perhaps my assumption that k = |a - b| is the same for both equations is incorrect? Wait, no, because both areas have the same denominator, which is |a - b|. Therefore, if we have [(d - c)^2]/|a - b| = 18 and [(c + d)^2]/|a - b| = 72, then dividing gives 4, leading to the same relation between c and d. Therefore, I think this is correct.

But let me check with different numbers. Suppose I take c = 3, d = 9, then (c + d) = 12, (d - c) = 6. 12^2 / 6^2 = 144 / 36 = 4. Correct. Then, if |a - b| = 2, then first area is 36 / 2 = 18, second is 144 / 2 = 72. Correct.

But maybe if there are other values of c and d where [(c + d)^2]/[(d - c)^2] = 4 but with smaller c and d? For example, if c = 1, d = 3. Then (1 + 3)^2 / (3 - 1)^2 = 16 / 4 = 4. Then, this would satisfy the ratio. Wait! Wait, this is case 2, where c = 3d? Wait no, here c = 1, d = 3. So d = 3c? No, d = 3*1 = 3, so d = 3c. Hence, case 1. Wait, if c = 1, d = 3, then |a - b| = [ (d - c)^2 ] / 18 = (2)^2 / 18 = 4 / 18 = 2/9. Which is not integer. Therefore, invalid. Therefore, even though the ratio is 4, the denominator |a - b| would not be integer. Therefore, this is impossible. Hence, why in our previous reasoning, we needed c to be multiple of 3.

So for example, if c = 3m, d = 9m, then (d - c) = 6m, (c + d) = 12m, then [(c + d)^2]/[(d - c)^2] = (144m²)/(36m²) = 4. Then |a - b| = (d - c)^2 / 18 = (36m²)/18 = 2m². Therefore, for m = 1, we get |a - b| = 2*1 = 2, which is integer. If m = 1, then c = 3, d = 9. If m = 2, c = 6, d = 18, |a - b| = 8, etc. Similarly for other m's.

Therefore, for m = 1, we get minimal sum. Therefore, 16?

But wait, let me check the problem statement again. It says "the parallelogram bounded by the lines y=ax+c, y=ax+d, y=bx+c, and y=bx+d has area 18". Similarly for the other. If we take a = 3, b = 1, c = 3, d = 9, then the first parallelogram is between y = 3x + 3, y = 3x + 9, y = x + 3, y = x + 9. Let me confirm the area with shoelace formula.

Vertices:

Intersection of y=3x+3 and y=x+3: 3x + 3 = x + 3 ⇒ 2x = 0 ⇒ x=0, y=3. So (0, 3).

Intersection of y=3x+3 and y=x+9: 3x + 3 = x + 9 ⇒ 2x = 6 ⇒ x = 3, y = 3*3 + 3 = 12. So (3, 12).

Intersection of y=3x+9 and y=x+9: 3x + 9 = x + 9 ⇒ 2x = 0 ⇒ x=0, y=9. So (0, 9).

Intersection of y=3x+9 and y=x+3: 3x + 9 = x + 3 ⇒ 2x = -6 ⇒ x = -3, y = -3 + 3 = 0. So (-3, 0).

Wait, but hold on, these points may not be in order. Let me list all four vertices:

(0,3), (3,12), (0,9), (-3,0). Wait, connecting these? Let me check coordinates again:

Wait, actually, the four lines are:

1. y = 3x + 3

2. y = 3x + 9

3. y = x + 3

4. y = x + 9

So intersections:

1 & 3: (0,3)

1 & 4: 3x + 3 = x + 9 ⇒ 2x = 6 ⇒ x = 3, y = 3*3 + 3 = 12 ⇒ (3,12)

2 & 3: 3x + 9 = x + 3 ⇒ 2x = -6 ⇒ x = -3, y = -3 + 3 = 0 ⇒ (-3, 0)

2 & 4: 3x + 9 = x + 9 ⇒ 2x = 0 ⇒ x = 0, y = 0 + 9 = 9 ⇒ (0,9)

So the four vertices are (0,3), (3,12), (0,9), (-3, 0). Let me order them properly for shoelace.

Let me arrange them in order. Let me think of the parallelogram. Starting at (0,3), moving along y=3x + 3 to (3,12)? Wait, no. Actually, the sides are between the lines. Maybe the correct order is (0,3), (3,12), (0,9), (-3,0), back to (0,3). Let me apply the shoelace formula:

List the coordinates in order:

(0, 3), (3, 12), (0, 9), (-3, 0), (0, 3)

Compute shoelace sum:

Sum1 = (0*12 + 3*9 + 0*0 + (-3)*3) = 0 + 27 + 0 -9 = 18

Sum2 = (3*3 + 12*0 + 9*(-3) + 0*0) = 9 + 0 -27 + 0 = -18

Area = 1/2 |Sum1 - Sum2| = 1/2 |18 - (-18)| = 1/2 * 36 = 18. Correct. So that works.

For the second parallelogram with lines y=3x+3, y=3x-9, y=x+3, y=x-9.

Let me compute its area. Similarly, 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 = -6 + 3 = -3 ⇒ (-6, -3)

Intersection of y=3x-9 and y=x+3: 3x -9 = x +3 ⇒ 2x = 12 ⇒ x = 6, y = 6 + 3 = 9 ⇒ (6, 9)

Intersection of y=3x-9 and y=x-9: 3x -9 = x -9 ⇒ 2x = 0 ⇒ x = 0, y = 0 -9 = -9 ⇒ (0, -9)

Vertices: (0,3), (-6, -3), (6,9), (0, -9). Let me order them properly. Maybe (0,3), (-6,-3), (0,-9), (6,9), back to (0,3). Let's apply shoelace.

Sum1 = 0*(-3) + (-6)*(-9) + 0*9 + 6*3 = 0 + 54 + 0 + 18 = 72

Sum2 = 3*(-6) + (-3)*0 + (-9)*6 + 9*0 = -18 + 0 -54 + 0 = -72

Area = 1/2 |72 - (-72)| = 1/2 * 144 = 72. Correct. So this also checks out. Therefore, the case with a=3, b=1, c=3, d=9 is valid and gives the total sum 3+1+3+9=16. But the problem says "the smallest possible value of a+b+c+d". However, maybe there is a configuration with smaller numbers?

Wait, let me think again. What if there are other solutions where (c + d)^2 / (d - c)^2 = 4, but with different relationships? For example, if c and d are not multiples of 3, but somehow fractions? But no, since c and d are positive integers.

Alternatively, maybe my formula for the area is incorrect? Let me check with another example. Suppose a=2, b=1, c=1, d=3. Then |a - b| = 1. Compute the area as |(d - c)^2 / (a - b)| = (3 -1)^2 / 1 = 4. Let me compute manually.

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

Vertices:

Intersection of 2x+1 and x+1: 2x +1 = x +1 ⇒ x=0, y=1 ⇒ (0,1)

Intersection of 2x+1 and x+3: 2x +1 = x +3 ⇒ x=2, y=5 ⇒ (2,5)

Intersection of 2x+3 and x+3: 2x +3 = x +3 ⇒ x=0, y=3 ⇒ (0,3)

Intersection of 2x+3 and x+1: 2x +3 = x +1 ⇒ x=-2, y=-2 +1 = -1 ⇒ (-2, -1)

Vertices: (0,1), (2,5), (0,3), (-2,-1). Shoelace formula:

Order: (0,1), (2,5), (0,3), (-2,-1), (0,1)

Sum1 = 0*5 + 2*3 + 0*(-1) + (-2)*1 = 0 + 6 + 0 -2 = 4

Sum2 = 1*2 + 5*0 + 3*(-2) + (-1)*0 = 2 + 0 -6 + 0 = -4

Area = 1/2 |4 - (-4)| = 4. Which matches the formula. Therefore, the formula is correct.

Therefore, in this case, with a=2, b=1, c=1, d=3, area is 4. If we use our previous equations:

For this case, (d - c)^2 / |a - b| = (2)^2 / 1 = 4, and (c + d)^2 / |a - b| = (4)^2 / 1 = 16. So if someone had 18 and 72, they would have to scale accordingly. But in our problem, we have the equations:

(d - c)^2 / k = 18

(c + d)^2 / k = 72

So if we have 18 and 72, then the ratio is 4, which is the same as the example. However, in this case, if we set (d - c)^2 / k = 18 and (c + d)^2 / k = 72, then (c + d)^2 / (d - c)^2 = 4, which as we saw, implies d = 3c or c = 3d. Therefore, in that case, k must be such that (d - c)^2 = 18k and (c + d)^2 = 72k. Which is the same as before. But in our previous example, k was 1. Therefore, if we want to have (d - c)^2 = 18k and (c + d)^2 = 72k, then (c + d)^2 = 4*(d - c)^2. Which is the same condition. Therefore, the same logic applies. Therefore, to find minimal a + b + c + d, we need to choose minimal c and d such that d = 3c or c = 3d, and k = (d - c)^2 / 18 is integer. Which for d = 3c, gives k = (2c)^2 / 18 = 4c² / 18 = 2c² / 9. Therefore, 2c² must be divisible by 9 ⇒ c must be divisible by 3. Let me let c = 3m ⇒ d = 9m. Then k = 2*(9m²)/9 = 2m². Therefore, |a - b| = 2m². To minimize a + b + c + d, we need minimal m, which is 1. Then, c = 3, d = 9, |a - b| = 2. Hence, the minimal a + b is 1 + 3 = 4 (if we take b = 1, a = 3). Therefore, total sum 4 + 3 + 9 = 16. Hence, 16 is the minimal.

But maybe there are other values where k is different? For example, suppose that (d - c)^2 = 18k and (c + d)^2 = 72k, but k is not necessarily equal to |a - b|. Wait, no, because we derived earlier that the area is |(d - c)^2 / (a - b)| = 18 and |(c + d)^2 / (a - b)| = 72, so k = |a - b|. Therefore, k must equal both (d - c)^2 / 18 and (c + d)^2 / 72. Therefore, these two expressions must be equal, which gives the ratio of 4. Hence, the same condition. Therefore, the minimal sum is 16.

But let me check with m = 1 and different a and b. For example, if |a - b| = 2, then possible a and b are (3,1), (4,2), (5,3), etc. For example, if we take a = 2, b = 0 (but b must be positive), so invalid. If a = 1, b = -1 (invalid). So only positive integers. Hence, the minimal a + b is 1 + 3 = 4. Hence, total sum 3 + 1 + 3 + 9 = 16.

But wait, maybe there is a case where the formula is different? For example, if the lines are not in the order I considered? For example, maybe the distance between the lines isn't |c - d| / sqrt(a² + 1), but something else?

Wait, but no, the distance between y = ax + c and y = ax + d is |c - d| / sqrt(a² + 1). Similarly for the other pair. Then the area of the parallelogram is the product of the distances divided by sin(theta), where theta is the angle between the two lines. Wait, but earlier I derived it as |(d - c)^2 / (a - b)|, but how is that related?

Alternatively, maybe I need to reconcile these two expressions. Let me think. The area can also be calculated as the distance between the first pair of lines multiplied by the distance between the second pair of lines divided by sin(theta), where theta is the angle between the two lines. Let me check that.

Distance between first pair: D1 = |c - d| / sqrt(a² + 1)

Distance between second pair: D2 = |c - d| / sqrt(b² + 1) ?

Wait no, first pair is y = ax + c and y = ax + d: distance is |c - d| / sqrt(a² + 1). The second pair is y = bx + c and y = bx + d: distance is |c - d| / sqrt(b² + 1). The angle theta between the two lines can be calculated as tan(theta) = |(m2 - m1)/(1 + m1m2)| = |(b - a)/(1 + ab)|. Then, the area of the parallelogram is D1 * D2 / sin(theta). Let me compute that:

Area = [ |c - d| / sqrt(a² + 1) ] * [ |c - d| / sqrt(b² + 1) ] / sin(theta). But this seems complicated. However, earlier we derived the area as |(d - c)^2 / (a - b)|. Let me check if these are equivalent.

Alternatively, since we have two expressions, let me consider our previous example with a=3, b=1, c=3, d=9. Let me compute using this formula.

D1 = |3 - 9| / sqrt(9 + 1) = 6 / sqrt(10)

D2 = |3 - 9| / sqrt(1 + 1) = 6 / sqrt(2)

sin(theta): tan(theta) = |(b - a)/(1 + ab)| = |(1 - 3)/(1 + 3*1)| = | -2 / 4 | = 1/2. Therefore, theta = arctan(1/2). So sin(theta) = 1 / sqrt(1 + 4) = 1 / sqrt(5). Therefore, Area = (6/sqrt(10)) * (6/sqrt(2)) / (1/sqrt(5)) ) = (36 / sqrt(20)) * sqrt(5) ) = 36 / (sqrt(20)/sqrt(5)) ) = 36 / sqrt(4) ) = 36 / 2 = 18. Which matches our previous result. Therefore, the formula using distances and sin(theta) gives the same result as the cross product formula.

Therefore, both formulas are equivalent. Therefore, the formula |(d - c)^2 / (a - b)| is correct.

Therefore, our previous reasoning is correct. Hence, the minimal possible value is 16. However, let me check again if there are other possibilities where k is different.

For example, if c and d are not multiples of 3, but the value of |a - b| is such that 2c² / 9 is integer. Let me suppose c is 3m, as before, but if c is not multiple of 3, say c = 6, then 2*(6)^2 /9 = 2*36/9 = 8, which is integer. But c=6 would give d=18, which would be larger. Hence, total sum would be higher. Therefore, m=1 gives minimal sum.

But wait, maybe there are other values for c and d not in the d=3c or c=3d relationship? For example, suppose that [(c + d)^2]/[(d - c)^2] = 4, but with different c and d. For instance, c=2, d=4. Then (c + d)/(d - c) = 6/2 = 3 ≠ 2. Doesn't satisfy. If c=4, d=6: (4 + 6)/(6 - 4) = 10/2 = 5 ≠ 2. Not. If c=1, d= 3: (1 + 3)/(3 - 1) = 4/2 = 2. Which satisfies. But then for this case, (d - c)^2 / k = 18 ⇒ 2² /k = 18 ⇒ k = 4 / 18 = 2/9, which is not integer. So invalid. Similarly, c=2, d= 6: (2 + 6)/(6 - 2) = 8/4 = 2. So this satisfies. Then check if k = (d - c)^2 / 18 = (4)^2 / 18 = 16 / 18 = 8/9 not integer. Not valid. c=4, d= 12: (4 + 12)/(12 - 4) = 16 / 8 = 2. Then (d - c)^2 / 18 = 8² / 18 = 64 / 18 = 32/9. Not integer. So only when c is multiple of 3, we can get integer k. Therefore, our previous conclusion holds.

Hence, the minimal possible value of a + b + c + d is 16. But wait, the problem says "positive integers", and in our case, all are positive. However, I want to check if there is a case where the sum is smaller. For example, let me think of other possibilities where the area formulas hold with different relations.

Suppose we have different values for k. For example, suppose that k divides both 18 and 72. But since 18 divides 72, k must divide 18. Wait, because (d - c)^2 / k = 18 and (c + d)^2 / k = 72 ⇒ k divides both 18 and 72. Therefore, k divides gcd(18, 72) = 18. Therefore, possible values of k are divisors of 18: 1, 2, 3, 6, 9, 18.

But from previous, k = (d - c)^2 / 18 must be equal to (c + d)^2 / 72. So let me denote:

Let me set k as a positive integer that divides 18, so possible k = 1, 2, 3, 6, 9, 18. Let me check for each possible k:

For each k, we have:

From first equation: (d - c)^2 = 18k

From second equation: (c + d)^2 = 72k

Therefore, (c + d)^2 = 4*(d - c)^2 ⇒ which gives us the same relation as before. So for each k, we need to solve (c + d)^2 = 4*(d - c)^2 ⇒ same as before, which gives d = 3c or c = 3d. Therefore, for each k, solving:

Case 1: d = 3c

Then, (d - c)^2 = (2c)^2 = 4c² = 18k ⇒ c² = (18k)/4 = (9k)/2

Similarly, since c must be integer, 9k must be divisible by 2 ⇒ k must be even. Since k divides 18, k can be 2, 6, 18.

If k = 2: c² = 9*2 / 2 = 9 ⇒ c = 3 ⇒ d = 9. Which is our previous case, leading to a + b + c + d = 16.

If k = 6: c² = 9*6 / 2 = 27 ⇒ c² = 27 ⇒ c is not integer.

If k = 18: c² = 9*18 / 2 = 81 ⇒ c = 9 ⇒ d = 27. Then, |a - b| = k = 18. Therefore, a and b are such that |a - b| = 18. To minimize a + b, take b = 1, a = 19. Then total sum is 19 + 1 + 9 + 27 = 56, which is larger than 16. Hence, this is worse.

Case 2: c = 3d

Then, (d - c)^2 = (d - 3d)^2 = (-2d)^2 = 4d² = 18k ⇒ d² = (18k)/4 = (9k)/2. Similarly, need 9k divisible by 2 ⇒ k even. So k = 2, 6, 18.

For k = 2: d² = 9*2 / 2 = 9 ⇒ d = 3 ⇒ c = 9. Which is same as previous case, leading to total sum 16.

For k = 6: d² = 9*6 / 2 = 27 ⇒ d not integer.

For k = 18: d² = 9*18 / 2 = 81 ⇒ d = 9 ⇒ c = 27. Then |a - b| = 18, leading to a + b + c + d = same as above, 56.

Hence, only when k = 2, we get valid integers c and d, leading to total sum 16. Therefore, 16 is indeed minimal.

Wait, but let me check if there are other k values if we don't restrict ourselves to d = 3c or c = 3d. However, we know that the equation (c + d)^2 = 4*(d - c)^2 only has those two solutions. Therefore, no other possibilities.

Therefore, the minimal possible value of a + b + c + d is 16. However, let me check if the problem specifies that a, b, c, d are positive integers, and in our case they are. However, the problem says "smallest possible value of a + b + c + d". Therefore, 16 is the answer?

But let me check if there is another possible k. For example, if k is not a divisor of 18? Wait, earlier I thought that since k divides 18 from (d - c)^2 = 18k, but actually, no. Wait, k is |a - b|, which is a positive integer. The equations are:

(d - c)^2 = 18k and (c + d)^2 = 72k. Therefore, since (c + d)^2 = 4*(d - c)^2, which is 4*18k = 72k, so it's consistent. Therefore, k can be any positive integer such that (d - c)^2 = 18k has integer solutions for d and c, with c and d positive integers, and similarly for (c + d)^2 = 72k. However, the key point is that for (d - c)^2 = 18k and (c + d)^2 = 72k, we must have that 18k is a perfect square, and 72k is also a perfect square.

Wait, hold on! This might be another way to look at it. Since (d - c)^2 = 18k ⇒ 18k must be a perfect square. Similarly, 72k must be a perfect square. Let me denote:

Let me let 18k = m² and 72k = n². Then, n² = 72k = 4*18k = 4m² ⇒ n = 2m. Therefore, the relationship between m and n is maintained. Therefore, 18k must be a perfect square. Therefore, prime factorization of 18k must have even exponents.

Since 18 = 2 * 3². Therefore, for 18k to be a perfect square, k must supply the missing factors to make all exponents even. Therefore, k must be of the form 2 * t², where t is a positive integer. Because:

18k = 2 * 3² * k. For this to be square, the exponent of 2 must be even, so k must have an odd exponent of 2. Therefore, k must contain a factor of 2, and other primes must be squares. Let me write k = 2 * t². Therefore, 18k = 18*2*t² = 36t² = (6t)², which is a perfect square. Then, 72k = 72*2*t² = 144t² = (12t)², which is also a perfect square. Therefore, k must be twice a square. Therefore, k = 2t², where t is a positive integer. Therefore, substituting back into (d - c)^2 = 18k = 18*2t² = 36t² ⇒ d - c = 6t. Similarly, (c + d)^2 = 72k = 72*2t² = 144t² ⇒ c + d = 12t. Therefore, we have:

c + d = 12t

d - c = 6t

Solving these equations:

Adding both equations:

2d = 18t ⇒ d = 9t

Subtracting the second equation from the first:

2c = 6t ⇒ c = 3t

Therefore, c = 3t and d = 9t. Therefore, this is the same result as before. Hence, this is the general solution. Therefore, c = 3t, d = 9t. Then, |a - b| = k = 2t².

Therefore, to minimize a + b + c + d, we need to choose the smallest possible t. Since t is a positive integer, the minimal t is 1. Therefore, c = 3, d = 9, and |a - b| = 2*1² = 2. To minimize a + b, choose a and b such that their difference is 2 and they are positive integers. The minimal sum occurs when b = 1 and a = 3, so a + b = 4. Hence, total sum is a + b + c + d = 3 + 1 + 3 + 9 = 16.

Therefore, this confirms our previous conclusion. Therefore, the smallest possible value is 16. However, the problem says "the smallest possible value of a + b + c + d". But I need to check if there are any constraints that I missed. For example, are there any restrictions on a and b? The problem states that a, b, c, d are positive integers. In our case, a = 3, b = 1, c = 3, d = 9, all positive integers. So this is valid.

But just to be thorough, let me check if there are other values of t which might allow smaller sums. For example, if t = 1/2, but t must be integer. Hence, no. Therefore, t = 1 is the minimum. Therefore, 16 is the minimal sum.

But let me check the problem statement again. It says the parallelogram bounded by the lines y=ax+c, y=ax+d, y=bx+c, and y=bx+d has area 18. Similarly for the other one. Is there any possibility that the values could be different if the lines are arranged differently? For example, maybe the lines could intersect in a different way? But no, because given the slopes and intercepts, the intersections are uniquely determined.

Therefore, I think 16 is correct. However, I feel uncertain because sometimes problems have trick answers or I might have missed something. Let me check if there's a configuration with smaller numbers. For example, take t = 1, sum 16. Suppose someone tried t = 1, but with different a and b. For example, if a - b = 2, and instead of a = 3, b = 1, maybe a = 2, b = 0. But b must be positive, so b = 0 is invalid. Or a = 4, b = 2. Then a + b = 6, and total sum 6 + 3 + 9 = 18, which is larger than 16. Similarly, if a = 1, b = -1, but b must be positive. So no. Therefore, no smaller sum.

Therefore, I think the answer is 16. However, I recall that in some cases, problems might have different configurations where the area formula differs, but according to our derivation and verification with examples, this seems correct.

Wait, but to be absolutely sure, let me check if there's a way to have smaller numbers by considering that the formula for area is |(d - c)^2 / (a - b)|, and maybe |a - b| is larger but allows smaller c and d. For example, suppose |a - b| is larger, but c and d are smaller such that (d - c)^2 / |a - b| = 18 and (c + d)^2 / |a - b| = 72. For instance, suppose |a - b| = 9, then (d - c)^2 = 18*9 = 162, which is not a perfect square. Similarly, |a - b| = 8: (d - c)^2 = 144 ⇒ d - c = 12, and (c + d)^2 = 72*8 = 576 ⇒ c + d = 24. Then solving d - c = 12 and c + d = 24 ⇒ d = 18, c = 6. Then, check if this is valid. So c = 6, d = 18, |a - b| = 8. Then, a + b + c + d = a + b + 6 + 18 = a + b + 24. To minimize a + b, with |a - b| = 8. Minimal sum of a + b occurs when a = 1 and b = 9 (since 1 + 9 = 10), or a = 9 and b = 1. Hence, total sum 10 + 24 = 34, which is larger than 16. Therefore, worse.

Alternatively, take |a - b| = 18. Then (d - c)^2 = 18*18 = 324 ⇒ d - c = 18, and (c + d)^2 = 72*18 = 1296 ⇒ c + d = 36. Solving: d = c + 18, so 2c + 18 = 36 ⇒ 2c = 18 ⇒ c = 9, d = 27. Then, a + b + c + d = a + b + 36. Minimal a + b is 1 + 19 = 20, total sum 56. Larger than 16.

Alternatively, take |a - b| = 3: (d - c)^2 = 18*3 = 54, not square. Similarly for |a - b| = 4: (d - c)^2 = 72, which is not square. |a - b| = 6: (d - c)^2 = 108, not square. |a - b| = 12: (d - c)^2 = 216, not square. Hence, no improvement.

Therefore, indeed, the minimal sum is 16.

But just to check once more, if there's another case where c + d and d - c are multiples that give smaller sums. For example, suppose we take t = 1, which gives c = 3, d = 9. If there's a smaller t? t must be positive integer, so t=1 is the smallest. Hence, no.

Therefore, I believe the correct answer is 16. But to make sure, let me check once more with different a and b. Suppose I take a = 1, b = -1. But b must be positive. Not allowed. a = 4, b = 2: sum a + b = 6. c = 3, d = 9. Total sum 6 + 3 + 9 = 18. Larger than 16. So no.

Yes, 16 is the minimal.

**Final Answer**
\boxed{16}

To solve the problem, we consider the two parallelograms defined by certain lines. Both parallelograms are constructed from lines with slopes $ a $ and $ b $, and intercepts involving $ c $ and $ d $. These lines generate parallelograms whose areas are given as 18 and 72 respectively. Our goal is to minimize the sum $ a + b + c + d $, where all variables are positive integers.

---

### Key Observations:

1. **Area of a Parallelogram Defined by Parallel Lines:**

   The area of a parallelogram formed by the lines
   $ y = ax + c $, $ y = ax + d $, $ y = bx + c $, and $ y = bx + d $
   can be computed using:
   $$
   \text{Area} = \frac{(d - c)^2}{|a - b|}
   $$

   Similarly, the second parallelogram (with lines $ y = ax + c $, $ y = ax - d $, $ y = bx + c $, and $ y = bx - d $) gives an area of:
   $$
   \text{Area} = \frac{(c + d)^2}{|a - b|}
   $$

2. **Setting Up Equations:**
   From the problem, we are given:
   $$
   \frac{(d - c)^2}{|a - b|} = 18 \quad \text{and} \quad \frac{(c + d)^2}{|a - b|} = 72
   $$
   Dividing these equations gives:
   $$
   \frac{(c + d)^2}{(d - c)^2} = 4 \Rightarrow \frac{c + d}{d - c} = 2
   $$
   Solving this ratio yields two cases:
   - $ d = 3c $
   - $ c = 3d $

3. **Choosing the Case:**
   We'll choose the case $ d = 3c $ (the symmetric case $ c = 3d $ will be identical in terms of minimization). Substituting $ d = 3c $ into the first area formula:
   $$
   \frac{(3c - c)^2}{|a - b|} = \frac{(2c)^2}{|a - b|} = \frac{4c^2}{|a - b|} = 18
   \Rightarrow |a - b| = \frac{4c^2}{18} = \frac{2c^2}{9}
   $$
   For this to be an integer, $ 2c^2 $ must be divisible by 9. The smallest such $ c $ that satisfies this is $ c = 3 $, so $ c = 3 $, $ d = 9 $, and:
   $$
   |a - b| = \frac{2 \cdot 3^2}{9} = \frac{18}{9} = 2
   $$

4. **Minimizing $ a + b + c + d $:**
   To minimize the total sum, we choose the smallest possible positive integers $ a $ and $ b $ such that $ |a - b| = 2 $. The pair $ a = 3 $, $ b = 1 $ gives the minimal sum $ a + b = 4 $.

   So the total sum is:
   $$
   a + b + c + d = 3 + 1 + 3 + 9 = \boxed{16}
   $$

---

### 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 $(a,b,c,d)=(1,3,3,9)$.