Explanation:
Okay, so I need to find the area of an equiangular octagon that has four sides of length 1 and four sides of length √2/2. The sides are arranged so that no two consecutive sides have the same length. Hmm, let me start by recalling what an equiangular octagon is.

An equiangular octagon means all its internal angles are equal. For a regular octagon, which is both equiangular and equilateral, each internal angle is 135 degrees. But this octagon isn't regular because the sides are of different lengths. However, since it's equiangular, all angles are still 135 degrees.

Now, the sides alternate between lengths 1 and √2/2, right? Wait, the problem says four sides of length 1 and four sides of length √2/2, arranged so that no two consecutive sides have the same length. So the sides must alternate between 1 and √2/2. Let me confirm that. If there are four sides of each length and no two consecutive sides are the same, then the sides must alternate between the two lengths. So the order would be 1, √2/2, 1, √2/2, and so on.

But wait, an octagon has eight sides. So starting with 1, then √2/2, then 1, √2/2, 1, √2/2, 1, √2/2. That gives four of each. So the sides alternate between 1 and √2/2.

Now, how do I find the area? For regular polygons, there are formulas, but since this isn't regular, I need another approach. Maybe I can think of it as a modified regular octagon, or perhaps decompose it into simpler shapes.

Alternatively, since it's equiangular, maybe I can model it on a coordinate system, using vectors or something. Let me think.

In an equiangular octagon, each internal angle is 135 degrees. If I can figure out the coordinates of the vertices, then I can use the shoelace formula to compute the area. That might be a way.

But how do I find the coordinates? Let me recall that in an equiangular polygon, even if it's not regular, the sides can be represented as vectors with angles between them. Since each internal angle is 135 degrees, the external angles would be 45 degrees, because the external angle is 180 - internal angle. Wait, for a regular polygon, the sum of external angles is 360 degrees. For an octagon, each external angle is 360/8 = 45 degrees. So even though the sides are not equal, the external angles are still 45 degrees? Wait, is that true?

Wait, in a regular polygon, the external angles add up to 360 degrees, and each external angle is 360/n. But for an equiangular polygon, regardless of side lengths, the external angles must still add up to 360 degrees. Therefore, each external angle is 360/8 = 45 degrees. Therefore, even though the sides are not equal, the direction of each side changes by 45 degrees from the previous one.

Therefore, if I model the octagon as a closed polygon, starting at a point, moving along a direction, then turning 45 degrees, then moving along the next side, etc., until I return to the starting point.

So, if I can model this as vectors, each subsequent side is rotated by 45 degrees from the previous one. Let me try to formalize this.

Let me denote the sides as vectors. Let me assume that the first side is along the positive x-axis. Let me denote the sides as vectors v1, v2, v3, ..., v8, each with magnitude either 1 or √2/2, alternating. The direction of each subsequent vector is rotated by 45 degrees from the previous one.

But since the octagon is closed, the sum of all vectors must be zero.

So, if I can write equations for the sum of the vectors, then I can solve for the side lengths? Wait, but the side lengths are already given as alternating 1 and √2/2. Wait, but maybe there's a scaling factor? Wait, no, the problem says that there are four sides of length 1 and four sides of length √2/2. So maybe the sides alternate between those lengths.

But if I model the vectors, each subsequent vector is rotated by 45 degrees. Let me think. Let me assign directions. Let me suppose that the first side is along the x-axis. Then each subsequent side is turned by 45 degrees. However, in a polygon, the direction of the turn depends on whether it's convex or concave. Since all internal angles are 135 degrees, which is less than 180, so it's convex. Therefore, each external angle is 45 degrees, meaning that each turn is 45 degrees to the left (assuming we traverse the polygon counterclockwise).

Therefore, the direction of each side is rotated by 45 degrees from the previous one.

So, if I denote the first side as vector v1, then the next side v2 is v1 rotated by 45 degrees, but scaled by the length of the second side. Wait, no. Wait, the direction is rotated by 45 degrees, but the length is different.

Wait, more accurately, each side is a vector with magnitude equal to the side length, and direction determined by the cumulative rotation.

Let me think of the direction of each side. Let me define the direction of the first side as angle 0 degrees (along the x-axis). Then, the second side is at angle 0 + 45 degrees? Wait, no. Wait, when you traverse the polygon, each turn is an external angle. So, if you have an internal angle of 135 degrees, then the external angle is 45 degrees. So, when moving along the perimeter, at each vertex, you turn by the external angle.

So, if you start going along the first side, then at the first vertex, you turn by 45 degrees to get to the direction of the second side. Therefore, the direction of the second side is the direction of the first side plus 45 degrees. Similarly, the third side is direction of second side plus 45 degrees, etc.

Therefore, the direction of the k-th side is (k-1)*45 degrees.

But since the octagon is closed, the sum of all the vectors must be zero.

So, if I can write each side as a vector in the complex plane, then sum them up and set the total to zero.

Let me try that. Let me represent each side as a complex number. Let me denote the sides as s1, s2, s3, ..., s8, where s1 = 1, s2 = √2/2, s3 = 1, s4 = √2/2, s5 = 1, s6 = √2/2, s7 = 1, s8 = √2/2.

But wait, the problem says that the sides are arranged so that no two consecutive sides have the same length. So the order is alternating between 1 and √2/2. So starting with 1, then √2/2, then 1, etc.

But if I model the directions as angles, then each side is at angle (k-1)*45 degrees.

But in complex plane, each side can be represented as length * e^{iθ}, where θ is the angle.

Therefore, the total displacement is sum_{k=1}^8 s_k * e^{iθ_k} = 0.

But θ_k is the angle of the k-th side. Since each subsequent side is rotated by 45 degrees from the previous one, starting from θ1 = 0 degrees.

Therefore, θ_k = (k - 1) * 45 degrees.

But converting degrees to radians for calculation purposes, but maybe I can work in degrees.

Alternatively, since complex numbers can be multiplied by e^{iπ/4} each time.

But maybe I can write equations for the real and imaginary parts.

Let me denote each side as a vector. Let me write them out:

s1: length 1, angle 0°
s2: length √2/2, angle 45°
s3: length 1, angle 90°
s4: length √2/2, angle 135°
s5: length 1, angle 180°
s6: length √2/2, angle 225°
s7: length 1, angle 270°
s8: length √2/2, angle 315°

Wait, but if we start at angle 0°, then each subsequent side is rotated by 45°, so yes, the angles are 0°, 45°, 90°, ..., 315°.

But then, if we add up all these vectors, they should sum to zero. However, if we just take the given side lengths, would that hold? Probably not, because the problem says that such an octagon exists. Therefore, maybe the side lengths are not exactly 1 and √2/2, but scaled? Wait, but the problem says that there are four sides of length 1 and four sides of length √2/2. So maybe the given side lengths are correct, but the problem is that when you add up the vectors, they don't close? Wait, but the problem says that such an octagon exists. Therefore, maybe my assumption about the angles is wrong?

Wait, perhaps the direction of the sides isn't just increasing by 45 degrees each time? Wait, no. Because in an equiangular polygon, the direction of the sides changes by the external angle each time. Since the internal angle is 135°, the external angle is 45°, so the direction of the sides changes by 45° each time.

But if I take the given side lengths, then the sum of the vectors would not be zero. Therefore, there must be some scaling factor? Wait, but the problem says that the sides are of lengths 1 and √2/2. Maybe my mistake is in the angles?

Wait, let me think again. If the octagon is equiangular, then the sides are not necessarily rotated by 45 degrees each time. Wait, no. Wait, in a regular octagon, each side is rotated by 45 degrees from the previous one. But in an equiangular octagon, even if sides are different lengths, the direction between sides is still determined by the external angles.

Wait, but if the external angles are all 45 degrees, then the direction of each side is rotated by 45 degrees from the previous one. Therefore, the angles of the sides are 0°, 45°, 90°, ..., 315°, as I thought.

But if I take the given side lengths, then the sum of the vectors would not be zero. Therefore, there must be some error in my reasoning.

Wait, maybe the problem is that the sides are not alternating between 1 and √2/2? Wait, the problem says four sides of length 1 and four sides of length √2/2, arranged so that no two consecutive sides have the same length. Therefore, they must alternate.

But if I take that, then the sum of the vectors would not be zero. Therefore, maybe the side lengths are not exactly 1 and √2/2, but scaled versions? Wait, but the problem says they are.

Wait, maybe I need to check. Let me try to compute the sum of the vectors.

Let me compute the real and imaginary parts.

First, convert each side to complex numbers.

s1: length 1, angle 0°: 1 * (cos 0° + i sin 0°) = 1 + 0i

s2: length √2/2, angle 45°: (√2/2)(cos 45° + i sin 45°) = (√2/2)(√2/2 + i√2/2) = ( (√2 * √2)/4 + i (√2 * √2)/4 ) = (2/4 + i 2/4) = 1/2 + i/2

s3: length 1, angle 90°: 1*(cos 90° + i sin 90°) = 0 + i*1 = 0 + i

s4: length √2/2, angle 135°: (√2/2)(cos 135° + i sin 135°) = (√2/2)(-√2/2 + i√2/2) = (-2/4 + i 2/4) = -1/2 + i/2

s5: length 1, angle 180°: 1*(cos 180° + i sin 180°) = -1 + 0i

s6: length √2/2, angle 225°: (√2/2)(cos 225° + i sin 225°) = (√2/2)(-√2/2 - i√2/2) = (-2/4 - i 2/4) = -1/2 - i/2

s7: length 1, angle 270°: 1*(cos 270° + i sin 270°) = 0 - i

s8: length √2/2, angle 315°: (√2/2)(cos 315° + i sin 315°) = (√2/2)(√2/2 - i√2/2) = (2/4 - i 2/4) = 1/2 - i/2

Now, let's add all these up:

Real parts:

s1: 1

s2: 1/2

s3: 0

s4: -1/2

s5: -1

s6: -1/2

s7: 0

s8: 1/2

Adding them up:

1 + 1/2 + 0 - 1/2 -1 -1/2 + 0 + 1/2

Let me compute step by step:

Start with 1.

Add 1/2: 1 + 1/2 = 3/2

Add 0: 3/2

Subtract 1/2: 3/2 - 1/2 = 1

Subtract 1: 1 - 1 = 0

Subtract 1/2: 0 - 1/2 = -1/2

Add 0: -1/2

Add 1/2: -1/2 + 1/2 = 0

So total real part is 0.

Imaginary parts:

s1: 0

s2: 1/2

s3: 1

s4: 1/2

s5: 0

s6: -1/2

s7: -1

s8: -1/2

Adding them up:

0 + 1/2 + 1 + 1/2 + 0 -1/2 -1 -1/2

Step by step:

Start with 0.

Add 1/2: 1/2

Add 1: 1/2 + 1 = 3/2

Add 1/2: 3/2 + 1/2 = 2

Add 0: 2

Subtract 1/2: 2 - 1/2 = 3/2

Subtract 1: 3/2 - 1 = 1/2

Subtract 1/2: 1/2 - 1/2 = 0

So total imaginary part is 0.

Wait, so the sum of all vectors is 0 + 0i. That means that the octagon closes! So my initial thought was correct. Therefore, such an octagon exists with sides alternating between 1 and √2/2, with angles as described.

Therefore, the coordinates can be calculated by starting at a point, say (0,0), and moving along each vector.

But to compute the area, maybe I can use the shoelace formula. However, to do that, I need coordinates of all the vertices.

Alternatively, since I have the vectors, maybe I can compute the area by breaking it into parts.

Alternatively, since the octagon is equiangular, maybe it can be considered as a regular octagon with some sides adjusted. But I need to think of another way.

Alternatively, since I have the vectors, I can compute the coordinates step by step. Let me try that.

Let me start at the origin (0,0). Let me denote each vertex as V0, V1, V2, ..., V8, where V8 = V0.

Starting at V0 = (0, 0).

First side is s1: from V0 to V1, which is vector (1, 0). So V1 = V0 + (1, 0) = (1, 0).

Second side is s2: from V1 to V2, which is vector (1/2, 1/2). So V2 = V1 + (1/2, 1/2) = (1 + 1/2, 0 + 1/2) = (3/2, 1/2).

Third side is s3: from V2 to V3, which is vector (0, 1). So V3 = V2 + (0, 1) = (3/2, 1/2 + 1) = (3/2, 3/2).

Fourth side is s4: from V3 to V4, which is vector (-1/2, 1/2). So V4 = V3 + (-1/2, 1/2) = (3/2 - 1/2, 3/2 + 1/2) = (1, 2).

Fifth side is s5: from V4 to V5, which is vector (-1, 0). So V5 = V4 + (-1, 0) = (1 - 1, 2 + 0) = (0, 2).

Sixth side is s6: from V5 to V6, which is vector (-1/2, -1/2). So V6 = V5 + (-1/2, -1/2) = (0 - 1/2, 2 - 1/2) = (-1/2, 3/2).

Seventh side is s7: from V6 to V7, which is vector (0, -1). So V7 = V6 + (0, -1) = (-1/2, 3/2 - 1) = (-1/2, 1/2).

Eighth side is s8: from V7 to V8, which is vector (1/2, -1/2). So V8 = V7 + (1/2, -1/2) = (-1/2 + 1/2, 1/2 - 1/2) = (0, 0).

Perfect! So the coordinates are:

V0: (0, 0)

V1: (1, 0)

V2: (3/2, 1/2)

V3: (3/2, 3/2)

V4: (1, 2)

V5: (0, 2)

V6: (-1/2, 3/2)

V7: (-1/2, 1/2)

V8: (0, 0)

Now, with these coordinates, I can apply the shoelace formula to compute the area.

The shoelace formula says that for a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn), the area is |1/2 * sum_{i=1 to n} (xi*yi+1 - xi+1*yi)|, where xn+1 = x1, yn+1 = y1.

So let me list all the coordinates:

V0: (0, 0)

V1: (1, 0)

V2: (3/2, 1/2)

V3: (3/2, 3/2)

V4: (1, 2)

V5: (0, 2)

V6: (-1/2, 3/2)

V7: (-1/2, 1/2)

V8: (0, 0)

So let me compute each term xi*yi+1 - xi+1*yi for i from 0 to 7.

Let me make a table:

i | xi | yi | xi+1 | yi+1 | xi*yi+1 - xi+1*yi
---|----|----|------|------|-------------------
0 | 0  | 0  | 1    | 0    | 0*0 - 1*0 = 0
1 | 1  | 0  | 3/2  | 1/2  | 1*(1/2) - (3/2)*0 = 1/2 - 0 = 1/2
2 | 3/2 | 1/2 | 3/2 | 3/2 | (3/2)*(3/2) - (3/2)*(1/2) = 9/4 - 3/4 = 6/4 = 3/2
3 | 3/2 | 3/2 | 1    | 2    | (3/2)*2 - 1*(3/2) = 3 - 3/2 = 3/2
4 | 1  | 2  | 0    | 2    | 1*2 - 0*2 = 2 - 0 = 2
5 | 0  | 2  | -1/2 | 3/2 | 0*(3/2) - (-1/2)*2 = 0 + 1 = 1
6 | -1/2 | 3/2 | -1/2 | 1/2 | (-1/2)*(1/2) - (-1/2)*(3/2) = -1/4 + 3/4 = 2/4 = 1/2
7 | -1/2 | 1/2 | 0    | 0    | (-1/2)*0 - 0*(1/2) = 0 - 0 = 0

Now, let me sum all these terms:

Term 0: 0

Term 1: 1/2

Term 2: 3/2

Term 3: 3/2

Term 4: 2

Term 5: 1

Term 6: 1/2

Term 7: 0

Adding them up:

Start with 0.

Add 1/2: 1/2

Add 3/2: 1/2 + 3/2 = 2

Add 3/2: 2 + 3/2 = 3.5 or 7/2

Add 2: 7/2 + 2 = 7/2 + 4/2 = 11/2

Add 1: 11/2 + 1 = 13/2

Add 1/2: 13/2 + 1/2 = 14/2 = 7

Add 0: 7

So the total sum is 7. Then the area is 1/2 * |7| = 7/2.

Wait, so the area is 7/2? That is 3.5? Let me check my calculations again because that seems straightforward, but let me verify each term again to make sure I didn't make a mistake.

Term 0: 0*0 - 1*0 = 0. Correct.

Term 1: 1*1/2 - 3/2*0 = 1/2. Correct.

Term 2: (3/2)*(3/2) - (3/2)*(1/2) = 9/4 - 3/4 = 6/4 = 3/2. Correct.

Term 3: (3/2)*2 - 1*(3/2) = 3 - 3/2 = 3/2. Correct.

Term 4: 1*2 - 0*2 = 2. Correct.

Term 5: 0*(3/2) - (-1/2)*2 = 0 + 1 = 1. Correct.

Term 6: (-1/2)*(1/2) - (-1/2)*(3/2) = -1/4 + 3/4 = 2/4 = 1/2. Correct.

Term 7: (-1/2)*0 - 0*(1/2) = 0. Correct.

Total sum: 0 + 1/2 + 3/2 + 3/2 + 2 + 1 + 1/2 + 0 = Let me add them again:

Start with 0.

Add 1/2: 1/2

Add 3/2: total 2

Add 3/2: total 3.5

Add 2: total 5.5

Add 1: total 6.5

Add 1/2: total 7.

Yes, so 7. Then area is 1/2 * 7 = 7/2.

But wait, 7/2 is 3.5. But let me think if that makes sense.

Alternatively, maybe I made a mistake in coordinates? Let me check coordinates again.

Starting from V0 (0,0).

V1: (1,0). Correct.

V2: (1 + 1/2, 0 + 1/2) = (3/2, 1/2). Correct.

V3: (3/2, 1/2 + 1) = (3/2, 3/2). Correct.

V4: (3/2 - 1/2, 3/2 + 1/2) = (1, 2). Correct.

V5: (1 - 1, 2 + 0) = (0, 2). Correct.

V6: (0 - 1/2, 2 - 1/2) = (-1/2, 3/2). Correct.

V7: (-1/2, 3/2 - 1) = (-1/2, 1/2). Correct.

V8: (-1/2 + 1/2, 1/2 - 1/2) = (0, 0). Correct.

Coordinates seem correct. Shoelace formula steps also seem correct. So area is 7/2?

But let me think about the shape. The octagon has coordinates from (0,0) to (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) to (0, 2) to (-1/2, 3/2) to (-1/2, 1/2) back to (0,0).

If I imagine this shape, it's symmetric? Let me check.

Looking at the coordinates, it's symmetric with respect to both the x-axis and y-axis? Wait, no. Because the points go from (0,0) to (1,0) to (3/2, 1/2)... but if I check for symmetry, maybe not. However, the area being 7/2 seems plausible?

Alternatively, maybe I can compute the area by dividing the octagon into simpler shapes.

Looking at the coordinates, maybe I can divide it into rectangles and triangles.

Alternatively, since the coordinates are known, maybe I can compute the area by breaking it into parts.

Alternatively, let me check with another method.

Alternatively, since the octagon can be considered as a larger square with smaller squares or rectangles cut out.

Looking at the coordinates, the maximum x-coordinate is 3/2, and the minimum is -1/2. Similarly, the maximum y-coordinate is 2, and the minimum is 0.

But maybe not. Alternatively, think of the octagon as a combination of trapezoids or something.

Alternatively, since I already have the shoelace formula giving 7/2, and the coordinates are correct, maybe that is the answer.

But let me check with another approach.

Another way to compute the area of an equiangular octagon is to use the formula for equiangular polygons. However, I don't recall such a formula off the top of my head.

Alternatively, since all angles are 135°, maybe we can model the octagon as a combination of rectangles and isosceles right triangles?

Wait, but given the side lengths, maybe not.

Alternatively, think of the octagon as a regular octagon with some sides adjusted.

But perhaps not.

Alternatively, since we have coordinates, maybe compute the area by dividing into triangles.

Alternatively, since the shoelace formula is straightforward and I verified the coordinates, maybe 7/2 is correct.

But let me check with another method.

Let me compute the area by breaking the octagon into parts.

Looking at the coordinates:

From (0,0) to (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) to (0, 2) to (-1/2, 3/2) to (-1/2, 1/2) to (0,0).

Let me divide this into simpler shapes.

First, notice that the figure is symmetric with respect to the line y = x? Wait, not sure. Let me check.

If I reflect over y = x, the point (1,0) would go to (0,1), but (0,1) is not a vertex. So maybe not symmetric.

Alternatively, maybe divide the octagon into a central rectangle and four trapezoids or something.

Alternatively, think of the octagon as a larger square with side length 2, but with some parts cut off.

Wait, looking at the coordinates, the maximum x is 3/2, and the minimum is -1/2. The maximum y is 2, and minimum is 0.

Alternatively, think of the octagon as a combination of a rectangle and two squares?

Alternatively, maybe compute the area by using coordinates.

Alternatively, since I already have the shoelace formula result, and the coordinates are correct, maybe I can accept that.

But let me check with another approach.

Let me compute the area using vectors. The area can also be calculated as half the magnitude of the sum over edges of (x_i y_{i+1} - x_{i+1} y_i), which is exactly what the shoelace formula does. So since I did that and got 7/2, maybe that's correct.

Alternatively, maybe I can compute the area by using the formula for equiangular polygons.

In general, for an equiangular polygon, even if it's not regular, there is a formula involving the side lengths and the angles. However, I don't remember it exactly.

Alternatively, since all angles are 135°, which is 180° - 45°, maybe we can model the octagon as a combination of vectors with angles 45° apart.

Alternatively, think of the octagon as a rectangle with side lengths a and b, with four isosceles right triangles cut off from each corner. Wait, but in a regular octagon, that's the case. However, in this case, the sides are not equal.

Wait, in a regular octagon, if you have a square and cut off the corners, each corner cut is an isosceles right triangle. However, in this case, the sides are of different lengths.

But maybe we can think of this octagon as a rectangle with some modifications.

Looking at the coordinates, the octagon has vertices at (0,0), (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), (0, 2), (-1/2, 3/2), (-1/2, 1/2), back to (0,0).

If I imagine this, the figure seems to have a sort of symmetry. Let me try to compute the area by breaking it into parts.

First, consider the rectangle from (-1/2, 1/2) to (3/2, 3/2). Wait, but that might not be helpful.

Alternatively, divide the octagon into the following parts:

1. The rectangle from (0,0) to (1,0) to (1,2) to (0,2) to (0,0). Wait, but that rectangle would have area 1*2 = 2. However, the octagon has more area.

But the octagon also includes the areas from (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) and similarly on the left side.

Alternatively, think of the octagon as a combination of a central rectangle and four trapezoids.

Alternatively, since I have coordinates, maybe use the shoelace formula again.

Alternatively, maybe I can check with vectors.

Alternatively, think of the octagon as a combination of squares and triangles.

But given that the shoelace formula gives 7/2, and I verified the coordinates, maybe that is correct.

But let me check with another coordinate system.

Alternatively, maybe I can compute the area by using the formula for the area of a polygon with sides and angles.

In general, for a polygon with sides a1, a2,..., an and angles θ1, θ2,..., θn, there is a formula, but I don't recall it.

Alternatively, since it's equiangular, all angles are equal, so maybe we can use some formula.

Alternatively, think of the octagon as a combination of vectors.

But since I already have the coordinates and the shoelace formula gives 7/2, which is 3.5, and the problem is from an Olympiad, maybe the answer is 7/2?

But let me check with another approach.

Let me compute the area by dividing the octagon into triangles.

For example, pick a point inside the octagon and connect it to all vertices. However, without knowing the exact position, it might be complex.

Alternatively, since we have coordinates, maybe divide the octagon into simpler polygons.

Looking at the coordinates:

From (0,0) to (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) to (0, 2) to (-1/2, 3/2) to (-1/2, 1/2) to (0,0).

Let me divide this into two trapezoids and a rectangle?

Alternatively, divide it into the following parts:

1. The rectangle from (0,0) to (1,0) to (1,2) to (0,2) to (0,0). Area = 1*2 = 2.

2. The area from (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2). This is a trapezoid? Or maybe a parallelogram?

Wait, the points (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2).

This shape is a parallelogram? Let me check. The vector from (1,0) to (3/2, 1/2) is (1/2, 1/2). The vector from (1,0) to (1, 2) is (0, 2). But the other sides: from (3/2, 1/2) to (3/2, 3/2) is (0, 1), and from (3/2, 3/2) to (1, 2) is (-1/2, 1/2).

Alternatively, maybe compute the area of this quadrilateral.

Using shoelace formula for this quadrilateral:

Coordinates: (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), back to (1,0).

Compute terms:

Term1: 1*(1/2) - 3/2*0 = 1/2 - 0 = 1/2

Term2: 3/2*(3/2) - 3/2*(1/2) = 9/4 - 3/4 = 6/4 = 3/2

Term3: 3/2*2 - 1*(3/2) = 3 - 3/2 = 3/2

Term4: 1*0 - 1*2 = 0 - 2 = -2

Sum: 1/2 + 3/2 + 3/2 - 2 = (1/2 + 3/2) + (3/2 - 2) = 2 + (-1/2) = 3/2. Then area is 1/2 * |3/2| = 3/4.

Wait, but that seems small. Wait, let me check again:

Wait, shoelace formula for quadrilateral:

List the coordinates:

(1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), (1,0)

Compute xi*yi+1 - xi+1*yi for each i:

i=0: x0=1, y0=0; x1=3/2, y1=1/2: 1*(1/2) - 3/2*0 = 1/2 - 0 = 1/2

i=1: x1=3/2, y1=1/2; x2=3/2, y2=3/2: 3/2*(3/2) - 3/2*(1/2) = 9/4 - 3/4 = 6/4 = 3/2

i=2: x2=3/2, y2=3/2; x3=1, y3=2: 3/2*2 - 1*(3/2) = 3 - 3/2 = 3/2

i=3: x3=1, y3=2; x4=1, y4=0: 1*0 - 1*2 = 0 - 2 = -2

Sum: 1/2 + 3/2 + 3/2 - 2 = (1/2 + 3/2 + 3/2) - 2 = ( (1 + 3 + 3)/2 ) - 2 = 7/2 - 2 = 7/2 - 4/2 = 3/2.

Then area is 1/2 * |3/2| = 3/4.

So the area of this quadrilateral is 3/4. Similarly, there is a symmetrical quadrilateral on the left side.

Similarly, the left side has points (-1/2, 3/2), (-1/2, 1/2), (0,0), (0, 2)? Wait, no. Wait, the left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2)? No, the octagon's left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0)? Wait, no.

Wait, the octagon's vertices are:

(0,0), (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), (0, 2), (-1/2, 3/2), (-1/2, 1/2), back to (0,0).

So the left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0). Wait, but that's a triangle?

Wait, maybe I need to consider symmetry. The area on the right side (the quadrilateral we calculated) is 3/4, and there is a symmetrical area on the left side.

Similarly, there are two more areas on the top and bottom?

Alternatively, maybe the entire area is the rectangle (area 2) plus two quadrilaterals (each 3/4) plus two triangles?

Wait, let me think again.

The octagon can be divided into:

1. The central rectangle from (0,0) to (1,0) to (1,2) to (0,2) to (0,0). Area = 2.

2. The two trapezoids on the right and left sides.

Wait, the right trapezoid is from (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2). Which we calculated as 3/4. Similarly, the left trapezoid would be from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2)? Wait, no.

Wait, the left side has points (-1/2, 3/2), (-1/2, 1/2), (0,0). But how does that connect?

Wait, the octagon's left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0). So that's a triangle?

Wait, if I consider the left side, from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2) to (1, 2) to (1, 0)...

Alternatively, maybe the left side has a similar area to the right side.

Let me compute the area of the left trapezoid.

The left trapezoid would be from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2) to (-1/2, 3/2). Wait, but that's not a trapezoid.

Alternatively, maybe the area on the left is a triangle and a rectangle?

Alternatively, maybe I need to compute the area of the left side.

Let me take the points (-1/2, 3/2), (-1/2, 1/2), (0,0), and (0, 2).

Wait, if I connect (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2) to (-1/2, 3/2).

This is a quadrilateral. Let me apply the shoelace formula to this quadrilateral.

Coordinates: (-1/2, 3/2), (-1/2, 1/2), (0, 0), (0, 2), back to (-1/2, 3/2).

Compute terms:

Term1: (-1/2)*(1/2) - (-1/2)*(3/2) = (-1/4) - (-3/4) = (-1/4 + 3/4) = 2/4 = 1/2

Term2: (-1/2)*0 - 0*(1/2) = 0 - 0 = 0

Term3: 0*2 - 0*0 = 0 - 0 = 0

Term4: 0*(3/2) - (-1/2)*2 = 0 - (-1) = 1

Sum: 1/2 + 0 + 0 + 1 = 3/2. Area is 1/2 * |3/2| = 3/4.

So the left side also has an area of 3/4.

Therefore, total area would be central rectangle (2) + right trapezoid (3/4) + left trapezoid (3/4) = 2 + 3/4 + 3/4 = 2 + 1.5 = 3.5 = 7/2.

Which matches the shoelace formula result.

Therefore, the area is indeed 7/2.

But let me check once more.

Alternatively, think of the entire figure as a combination of the central rectangle (area 2) and two trapezoids (each 3/4), totaling 2 + 3/4 + 3/4 = 3.5.

Alternatively, maybe there are more areas?

Wait, the octagon also has the top and bottom parts?

Wait, no, because the central rectangle already includes the area from y=0 to y=2. The trapezoids on the sides are the extensions on the right and left.

Alternatively, maybe there are also areas on the top and bottom?

Wait, looking at the coordinates, the octagon's topmost point is at y=2, and the bottommost is at y=0. So the central rectangle already covers from y=0 to y=2.

Therefore, the total area is indeed 7/2.

Therefore, I think the answer is 7/2.

But let me check with another method.

Alternatively, since the octagon can be considered as a regular octagon with some sides adjusted.

But I don't know the side lengths of a regular octagon. However, maybe I can compute the area of the regular octagon and adjust it?

But since the given octagon is not regular, this might not be straightforward.

Alternatively, think of the octagon as a combination of squares and rectangles.

But given that I have two different methods giving me 7/2, I think that is correct.

Therefore, the area of the octagon is 7/2, which is 3.5.

But let me check if the problem says "four sides of length 1 and four sides of length √2/2". In my calculation, the sides are alternating between 1 and √2/2, which matches the problem's condition.

Therefore, I think the answer is 7/2.

But just to be thorough, let me check if there's any possible mistake in the shoelace formula.

The shoelace formula is reliable if the coordinates are correct. Since I walked through the coordinates step by step and they close properly, and the shoelace formula gives 7/2, which is confirmed by breaking into parts, I think it's safe.

Therefore, the area of the octagon is 7/2.

**Final Answer**
\boxed{\dfrac{7}{2}}

To find the area of the given equiangular octagon, we begin by understanding its structure and properties.

---

### **Understanding the Octagon**

An **equiangular octagon** has all internal angles equal. For an octagon, each internal angle is:

$$
\text{Internal angle} = \frac{(8 - 2) \cdot 180^\circ}{8} = 135^\circ
$$

This means that the **external angle** at each vertex is:

$$
\text{External angle} = 180^\circ - 135^\circ = 45^\circ
$$

Since the octagon is equiangular, the direction of each side changes by 45° from the previous one. The side lengths alternate between 1 and $\frac{\sqrt{2}}{2}$, and no two consecutive sides have the same length. This implies the sides alternate in the order:

$$
1, \frac{\sqrt{2}}{2}, 1, \frac{\sqrt{2}}{2}, \dots
$$

---

### **Modeling the Octagon as Vectors**

We can model the octagon as a closed polygon in the complex plane, where each side is a vector with a specific length and direction. Starting from the origin, we can compute the coordinates of each vertex by summing the vectors.

Let’s define the direction of each side as follows:

- Side 1: length 1, angle 0°
- Side 2: length $\frac{\sqrt{2}}{2}$, angle 45°
- Side 3: length 1, angle 90°
- Side 4: length $\frac{\sqrt{2}}{2}$, angle 135°
- Side 5: length 1, angle 180°
- Side 6: length $\frac{\sqrt{2}}{2}$, angle 225°
- Side 7: length 1, angle 270°
- Side 8: length $\frac{\sqrt{2}}{2}$, angle 315°

We convert each side into a vector in the complex plane and sum them up. The result is zero, confirming that the octagon is closed.

---

### **Computing the Area Using the Shoelace Formula**

We now compute the coordinates of the vertices by summing the vectors step by step. The resulting coordinates are:

- $V_0 = (0, 0)$
- $V_1 = (1, 0)$
- $V_2 = \left(\frac{3}{2}, \frac{1}{2}\right)$
- $V_3 = \left(\frac{3}{2}, \frac{3}{2}\right)$
- $V_4 = (1, 2)$
- $V_5 = (0, 2)$
- $V_6 = \left(-\frac{1}{2}, \frac{3}{2}\right)$
- $V_7 = \left(-\frac{1}{2}, \frac{1}{2}\right)$
- $V_8 = (0, 0)$

Using the **shoelace formula**:

$$
\text{Area} = \frac{1}{2} \left| \sum_{i=0}^{7} (x_i y_{i+1} - x_{i+1} y_i) \right|
$$

We compute each term and sum them up:

$$
\text{Sum} = 0 + \frac{1}{2} + \frac{3}{2} + \frac{3}{2} + 2 + 1 + \frac{1}{2} + 0 = 7
$$

$$
\text{Area} = \frac{1}{2} \cdot 7 = \frac{7}{2}
$$

---

### **Conclusion**

After verifying the coordinates and applying the shoelace formula, we find that the area of the equiangular octagon with alternating side lengths of 1 and $\frac{\sqrt{2}}{2}$ is:

$$
\boxed{\dfrac{7}{2}}
$$

[Assistant]
Explanation:
Okay, so I need to find the area of an equiangular octagon that has four sides of length 1 and four sides of length √2/2. The sides are arranged so that no two consecutive sides have the same length. Hmm, let me start by recalling what an equiangular octagon is.

An equiangular octagon means all its internal angles are equal. For a regular octagon, which is both equiangular and equilateral, each internal angle is 135 degrees. But this octagon isn't regular because the sides are of different lengths. However, since it's equiangular, all angles are still 135 degrees.

Now, the sides alternate between lengths 1 and √2/2, right? Wait, the problem says four sides of length 1 and four sides of length √2/2, arranged so that no two consecutive sides have the same length. So the sides must alternate between 1 and √2/2. Let me confirm that. If there are four sides of each length and no two consecutive sides are the same, then the sides must alternate between the two lengths. So the order would be 1, √2/2, 1, √2/2, and so on.

But wait, an octagon has eight sides. So starting with 1, then √2/2, then 1, √2/2, 1, √2/2, 1, √2/2. That gives four of each. So the sides alternate between 1 and √2/2.

Now, how do I find the area? For regular polygons, there are formulas, but since this isn't regular, I need another approach. Maybe I can think of it as a modified regular octagon, or perhaps decompose it into simpler shapes.

Alternatively, since it's equiangular, maybe I can model it on a coordinate system, using vectors or something. Let me think.

In an equiangular octagon, each internal angle is 135 degrees. If I can figure out the coordinates of the vertices, then I can use the shoelace formula to compute the area. That might be a way.

But how do I find the coordinates? Let me recall that in an equiangular polygon, even if it's not regular, the sides can be represented as vectors with angles between them. Since each internal angle is 135 degrees, the external angles would be 45 degrees, because the external angle is 180 - internal angle. Wait, for a regular polygon, the sum of external angles is 360 degrees. For an octagon, each external angle is 360/8 = 45 degrees. So even though the sides are not equal, the external angles are still 45 degrees? Wait, is that true?

Wait, in a regular polygon, the external angles add up to 360 degrees, and each external angle is 360/n. But for an equiangular polygon, regardless of side lengths, the external angles must still add up to 360 degrees. Therefore, each external angle is 360/8 = 45 degrees. Therefore, even though the sides are not equal, the direction of each side changes by 45 degrees from the previous one.

Therefore, if I model the octagon as a closed polygon, starting at a point, moving along a direction, then turning 45 degrees, then moving along the next side, etc., until I return to the starting point.

So, if I can model this as vectors, each subsequent side is rotated by 45 degrees from the previous one. Let me try to formalize this.

Let me denote the sides as vectors. Let me assume that the first side is along the positive x-axis. Let me denote the sides as vectors v1, v2, v3, ..., v8, each with magnitude either 1 or √2/2, alternating. The direction of each subsequent vector is rotated by 45 degrees from the previous one.

But since the octagon is closed, the sum of all vectors must be zero.

So, if I can write equations for the sum of the vectors, then I can solve for the side lengths? Wait, but the side lengths are already given as alternating 1 and √2/2. Wait, but maybe there's a scaling factor? Wait, no, the problem says that there are four sides of length 1 and four sides of length √2/2. So maybe the sides alternate between those lengths.

But if I model the vectors, each subsequent vector is rotated by 45 degrees. Let me think. Let me assign directions. Let me suppose that the first side is along the x-axis. Then each subsequent side is turned by 45 degrees. However, in a polygon, the direction of the turn depends on whether it's convex or concave. Since all internal angles are 135 degrees, which is less than 180, so it's convex. Therefore, each external angle is 45 degrees, meaning that each turn is 45 degrees to the left (assuming we traverse the polygon counterclockwise).

Therefore, the direction of each side is rotated by 45 degrees from the previous one.

So, if I denote the first side as vector v1, then the next side v2 is v1 rotated by 45 degrees, but scaled by the length of the second side. Wait, no. Wait, the direction is rotated by 45 degrees, but the length is different.

Wait, more accurately, each side is a vector with magnitude equal to the side length, and direction determined by the cumulative rotation.

Let me think of the direction of each side. Let me define the direction of the first side as angle 0 degrees (along the x-axis). Then, the second side is at angle 0 + 45 degrees? Wait, no. Wait, when you traverse the polygon, each turn is an external angle. So, if you have an internal angle of 135 degrees, then the external angle is 45 degrees. So, when moving along the perimeter, at each vertex, you turn by the external angle.

So, if you start going along the first side, then at the first vertex, you turn by 45 degrees to get to the direction of the second side. Therefore, the direction of the second side is the direction of the first side plus 45 degrees. Similarly, the third side is direction of second side plus 45 degrees, etc.

Therefore, the direction of the k-th side is (k-1)*45 degrees.

But since the octagon is closed, the sum of all the vectors must be zero.

So, if I can write each side as a vector in the complex plane, then sum them up and set the total to zero.

Let me try that. Let me represent each side as a complex number. Let me denote the sides as s1, s2, s3, ..., s8, where s1 = 1, s2 = √2/2, s3 = 1, s4 = √2/2, s5 = 1, s6 = √2/2, s7 = 1, s8 = √2/2.

But wait, the problem says that the sides are arranged so that no two consecutive sides have the same length. So the order is alternating between 1 and √2/2. So starting with 1, then √2/2, then 1, etc.

But if I model the directions as angles, then each side is at angle (k-1)*45 degrees.

But in complex plane, each side can be represented as length * e^{iθ}, where θ is the angle.

Therefore, the total displacement is sum_{k=1}^8 s_k * e^{iθ_k} = 0.

But θ_k is the angle of the k-th side. Since each subsequent side is rotated by 45 degrees from the previous one, starting from θ1 = 0 degrees.

Therefore, θ_k = (k - 1) * 45 degrees.

But converting degrees to radians for calculation purposes, but maybe I can work in degrees.

Alternatively, since complex numbers can be multiplied by e^{iπ/4} each time.

But maybe I can write equations for the real and imaginary parts.

Let me denote each side as a vector. Let me write them out:

s1: length 1, angle 0°
s2: length √2/2, angle 45°
s3: length 1, angle 90°
s4: length √2/2, angle 135°
s5: length 1, angle 180°
s6: length √2/2, angle 225°
s7: length 1, angle 270°
s8: length √2/2, angle 315°

Wait, but if we start at angle 0°, then each subsequent side is rotated by 45°, so yes, the angles are 0°, 45°, 90°, ..., 315°.

But then, if we add up all these vectors, they should sum to zero. However, if we just take the given side lengths, would that hold? Probably not, because the problem says that such an octagon exists. Therefore, maybe the side lengths are not exactly 1 and √2/2, but scaled? Wait, but the problem says that there are four sides of length 1 and four sides of length √2/2. So maybe the given side lengths are correct, but the problem is that when you add up the vectors, they don't close? Wait, but the problem says that such an octagon exists. Therefore, maybe my assumption about the angles is wrong?

Wait, perhaps the direction of the sides isn't just increasing by 45 degrees each time? Wait, no. Because in an equiangular polygon, the direction of the sides changes by the external angle each time. Since the internal angle is 135°, the external angle is 45°, so the direction of the sides changes by 45° each time.

But if I take the given side lengths, then the sum of the vectors would not be zero. Therefore, there must be some scaling factor? Wait, but the problem says that the sides are of lengths 1 and √2/2. Maybe my mistake is in the angles?

Wait, let me think again. If the octagon is equiangular, then the sides are not necessarily rotated by 45 degrees each time. Wait, no. Wait, in a regular octagon, each side is rotated by 45 degrees from the previous one. But in an equiangular octagon, even if sides are different lengths, the direction between sides is still determined by the external angles.

Wait, but if the external angles are all 45 degrees, then the direction of each side is rotated by 45 degrees from the previous one. Therefore, the angles of the sides are 0°, 45°, 90°, ..., 315°, as I thought.

But if I take the given side lengths, then the sum of the vectors would not be zero. Therefore, there must be some error in my reasoning.

Wait, maybe the problem is that the sides are not alternating between 1 and √2/2? Wait, the problem says four sides of length 1 and four sides of length √2/2, arranged so that no two consecutive sides have the same length. Therefore, they must alternate.

But if I take that, then the sum of the vectors would not be zero. Therefore, maybe the side lengths are not exactly 1 and √2/2, but scaled versions? Wait, but the problem says they are.

Wait, maybe I need to check. Let me try to compute the sum of the vectors.

Let me compute the real and imaginary parts.

First, convert each side to complex numbers.

s1: length 1, angle 0°: 1 * (cos 0° + i sin 0°) = 1 + 0i

s2: length √2/2, angle 45°: (√2/2)(cos 45° + i sin 45°) = (√2/2)(√2/2 + i√2/2) = ( (√2 * √2)/4 + i (√2 * √2)/4 ) = (2/4 + i 2/4) = 1/2 + i/2

s3: length 1, angle 90°: 1*(cos 90° + i sin 90°) = 0 + i*1 = 0 + i

s4: length √2/2, angle 135°: (√2/2)(cos 135° + i sin 135°) = (√2/2)(-√2/2 + i√2/2) = (-2/4 + i 2/4) = -1/2 + i/2

s5: length 1, angle 180°: 1*(cos 180° + i sin 180°) = -1 + 0i

s6: length √2/2, angle 225°: (√2/2)(cos 225° + i sin 225°) = (√2/2)(-√2/2 - i√2/2) = (-2/4 - i 2/4) = -1/2 - i/2

s7: length 1, angle 270°: 1*(cos 270° + i sin 270°) = 0 - i

s8: length √2/2, angle 315°: (√2/2)(cos 315° + i sin 315°) = (√2/2)(√2/2 - i√2/2) = (2/4 - i 2/4) = 1/2 - i/2

Now, let's add all these up:

Real parts:

s1: 1

s2: 1/2

s3: 0

s4: -1/2

s5: -1

s6: -1/2

s7: 0

s8: 1/2

Adding them up:

1 + 1/2 + 0 - 1/2 -1 -1/2 + 0 + 1/2

Let me compute step by step:

Start with 1.

Add 1/2: 1 + 1/2 = 3/2

Add 0: 3/2

Subtract 1/2: 3/2 - 1/2 = 1

Subtract 1: 1 - 1 = 0

Subtract 1/2: 0 - 1/2 = -1/2

Add 0: -1/2

Add 1/2: -1/2 + 1/2 = 0

So total real part is 0.

Imaginary parts:

s1: 0

s2: 1/2

s3: 1

s4: 1/2

s5: 0

s6: -1/2

s7: -1

s8: -1/2

Adding them up:

0 + 1/2 + 1 + 1/2 + 0 -1/2 -1 -1/2

Step by step:

Start with 0.

Add 1/2: 1/2

Add 1: 1/2 + 1 = 3/2

Add 1/2: 3/2 + 1/2 = 2

Add 0: 2

Subtract 1/2: 2 - 1/2 = 3/2

Subtract 1: 3/2 - 1 = 1/2

Subtract 1/2: 1/2 - 1/2 = 0

So total imaginary part is 0.

Wait, so the sum of all vectors is 0 + 0i. That means that the octagon closes! So my initial thought was correct. Therefore, such an octagon exists with sides alternating between 1 and √2/2, with angles as described.

Therefore, the coordinates can be calculated by starting at a point, say (0,0), and moving along each vector.

But to compute the area, maybe I can use the shoelace formula. However, to do that, I need coordinates of all the vertices.

Alternatively, since I have the vectors, maybe I can compute the area by breaking it into parts.

Alternatively, since the octagon is equiangular, maybe it can be considered as a regular octagon with some sides adjusted. But I need to think of another way.

Alternatively, since I have the vectors, I can compute the coordinates step by step. Let me try that.

Let me start at the origin (0,0). Let me denote each vertex as V0, V1, V2, ..., V8, where V8 = V0.

Starting at V0 = (0, 0).

First side is s1: from V0 to V1, which is vector (1, 0). So V1 = V0 + (1, 0) = (1, 0).

Second side is s2: from V1 to V2, which is vector (1/2, 1/2). So V2 = V1 + (1/2, 1/2) = (1 + 1/2, 0 + 1/2) = (3/2, 1/2).

Third side is s3: from V2 to V3, which is vector (0, 1). So V3 = V2 + (0, 1) = (3/2, 1/2 + 1) = (3/2, 3/2).

Fourth side is s4: from V3 to V4, which is vector (-1/2, 1/2). So V4 = V3 + (-1/2, 1/2) = (3/2 - 1/2, 3/2 + 1/2) = (1, 2).

Fifth side is s5: from V4 to V5, which is vector (-1, 0). So V5 = V4 + (-1, 0) = (1 - 1, 2 + 0) = (0, 2).

Sixth side is s6: from V5 to V6, which is vector (-1/2, -1/2). So V6 = V5 + (-1/2, -1/2) = (0 - 1/2, 2 - 1/2) = (-1/2, 3/2).

Seventh side is s7: from V6 to V7, which is vector (0, -1). So V7 = V6 + (0, -1) = (-1/2, 3/2 - 1) = (-1/2, 1/2).

Eighth side is s8: from V7 to V8, which is vector (1/2, -1/2). So V8 = V7 + (1/2, -1/2) = (-1/2 + 1/2, 1/2 - 1/2) = (0, 0).

Perfect! So the coordinates are:

V0: (0, 0)

V1: (1, 0)

V2: (3/2, 1/2)

V3: (3/2, 3/2)

V4: (1, 2)

V5: (0, 2)

V6: (-1/2, 3/2)

V7: (-1/2, 1/2)

V8: (0, 0)

Now, with these coordinates, I can apply the shoelace formula to compute the area.

The shoelace formula says that for a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn), the area is |1/2 * sum_{i=1 to n} (xi*yi+1 - xi+1*yi)|, where xn+1 = x1, yn+1 = y1.

So let me list all the coordinates:

V0: (0, 0)

V1: (1, 0)

V2: (3/2, 1/2)

V3: (3/2, 3/2)

V4: (1, 2)

V5: (0, 2)

V6: (-1/2, 3/2)

V7: (-1/2, 1/2)

V8: (0, 0)

So let me compute each term xi*yi+1 - xi+1*yi for i from 0 to 7.

Let me make a table:

i | xi | yi | xi+1 | yi+1 | xi*yi+1 - xi+1*yi
---|----|----|------|------|-------------------
0 | 0  | 0  | 1    | 0    | 0*0 - 1*0 = 0
1 | 1  | 0  | 3/2  | 1/2  | 1*(1/2) - (3/2)*0 = 1/2 - 0 = 1/2
2 | 3/2 | 1/2 | 3/2 | 3/2 | (3/2)*(3/2) - (3/2)*(1/2) = 9/4 - 3/4 = 6/4 = 3/2
3 | 3/2 | 3/2 | 1    | 2    | (3/2)*2 - 1*(3/2) = 3 - 3/2 = 3/2
4 | 1  | 2  | 0    | 2    | 1*2 - 0*2 = 2 - 0 = 2
5 | 0  | 2  | -1/2 | 3/2 | 0*(3/2) - (-1/2)*2 = 0 + 1 = 1
6 | -1/2 | 3/2 | -1/2 | 1/2 | (-1/2)*(1/2) - (-1/2)*(3/2) = -1/4 + 3/4 = 2/4 = 1/2
7 | -1/2 | 1/2 | 0    | 0    | (-1/2)*0 - 0*(1/2) = 0 - 0 = 0

Now, let me sum all these terms:

Term 0: 0

Term 1: 1/2

Term 2: 3/2

Term 3: 3/2

Term 4: 2

Term 5: 1

Term 6: 1/2

Term 7: 0

Adding them up:

Start with 0.

Add 1/2: 1/2

Add 3/2: 1/2 + 3/2 = 2

Add 3/2: 2 + 3/2 = 3.5 or 7/2

Add 2: 7/2 + 2 = 7/2 + 4/2 = 11/2

Add 1: 11/2 + 1 = 13/2

Add 1/2: 13/2 + 1/2 = 14/2 = 7

Add 0: 7

So the total sum is 7. Then the area is 1/2 * |7| = 7/2.

Wait, so the area is 7/2? That is 3.5? Let me check my calculations again because that seems straightforward, but let me verify each term again to make sure I didn't make a mistake.

Term 0: 0*0 - 1*0 = 0. Correct.

Term 1: 1*1/2 - 3/2*0 = 1/2. Correct.

Term 2: (3/2)*(3/2) - (3/2)*(1/2) = 9/4 - 3/4 = 6/4 = 3/2. Correct.

Term 3: (3/2)*2 - 1*(3/2) = 3 - 3/2 = 3/2. Correct.

Term 4: 1*2 - 0*2 = 2. Correct.

Term 5: 0*(3/2) - (-1/2)*2 = 0 + 1 = 1. Correct.

Term 6: (-1/2)*(1/2) - (-1/2)*(3/2) = -1/4 + 3/4 = 2/4 = 1/2. Correct.

Term 7: (-1/2)*0 - 0*(1/2) = 0. Correct.

Total sum: 0 + 1/2 + 3/2 + 3/2 + 2 + 1 + 1/2 + 0 = Let me add them again:

Start with 0.

Add 1/2: 1/2

Add 3/2: total 2

Add 3/2: total 3.5

Add 2: total 5.5

Add 1: total 6.5

Add 1/2: total 7.

Yes, so 7. Then area is 1/2 * 7 = 7/2.

But wait, 7/2 is 3.5. But let me think if that makes sense.

Alternatively, maybe I made a mistake in coordinates? Let me check coordinates again.

Starting from V0 (0,0).

V1: (1,0). Correct.

V2: (1 + 1/2, 0 + 1/2) = (3/2, 1/2). Correct.

V3: (3/2, 1/2 + 1) = (3/2, 3/2). Correct.

V4: (3/2 - 1/2, 3/2 + 1/2) = (1, 2). Correct.

V5: (1 - 1, 2 + 0) = (0, 2). Correct.

V6: (0 - 1/2, 2 - 1/2) = (-1/2, 3/2). Correct.

V7: (-1/2, 3/2 - 1) = (-1/2, 1/2). Correct.

V8: (-1/2 + 1/2, 1/2 - 1/2) = (0, 0). Correct.

Coordinates seem correct. Shoelace formula steps also seem correct. So area is 7/2?

But let me think about the shape. The octagon has coordinates from (0,0) to (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) to (0, 2) to (-1/2, 3/2) to (-1/2, 1/2) back to (0,0).

If I imagine this shape, it's symmetric? Let me check.

Looking at the coordinates, it's symmetric with respect to both the x-axis and y-axis? Wait, no. Because the points go from (0,0) to (1,0) to (3/2, 1/2)... but if I check for symmetry, maybe not. However, the area being 7/2 seems plausible?

Alternatively, maybe I can compute the area by dividing the octagon into simpler shapes.

Looking at the coordinates, maybe I can divide it into rectangles and triangles.

Alternatively, since the coordinates are known, maybe I can compute the area by breaking it into parts.

Alternatively, let me check with another method.

Alternatively, since the octagon can be considered as a larger square with smaller squares or rectangles cut out.

Looking at the coordinates, the maximum x-coordinate is 3/2, and the minimum is -1/2. Similarly, the maximum y-coordinate is 2, and the minimum is 0.

But maybe not. Alternatively, think of the octagon as a combination of trapezoids or something.

Alternatively, since I already have the shoelace formula giving 7/2, and the coordinates are correct, maybe that is the answer.

But let me check with another approach.

Another way to compute the area of an equiangular octagon is to use the formula for equiangular polygons. However, I don't recall such a formula off the top of my head.

Alternatively, since all angles are 135°, maybe we can model the octagon as a combination of rectangles and isosceles right triangles?

Wait, but given the side lengths, maybe not.

Alternatively, think of the octagon as a regular octagon with some sides adjusted.

But perhaps not.

Alternatively, since we have coordinates, maybe compute the area by dividing into triangles.

Alternatively, since the shoelace formula is straightforward and I verified the coordinates, maybe 7/2 is correct.

But let me check with another method.

Let me compute the area by breaking the octagon into parts.

Looking at the coordinates:

From (0,0) to (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) to (0, 2) to (-1/2, 3/2) to (-1/2, 1/2) to (0,0).

Let me divide this into simpler shapes.

First, notice that the figure is symmetric with respect to the line y = x? Wait, not sure. Let me check.

If I reflect over y = x, the point (1,0) would go to (0,1), but (0,1) is not a vertex. So maybe not symmetric.

Alternatively, maybe divide the octagon into a central rectangle and four trapezoids or something.

Alternatively, think of the octagon as a larger square with side length 2, but with some parts cut off.

Wait, looking at the coordinates, the maximum x is 3/2, and the minimum is -1/2. The maximum y is 2, and minimum is 0.

Alternatively, think of the octagon as a combination of a rectangle and two squares?

Alternatively, maybe compute the area by using coordinates.

Alternatively, since I already have the shoelace formula result, and the coordinates are correct, maybe I can accept that.

But let me check with another approach.

Let me compute the area using vectors. The area can also be calculated as half the magnitude of the sum over edges of (x_i y_{i+1} - x_{i+1} y_i), which is exactly what the shoelace formula does. So since I did that and got 7/2, maybe that's correct.

Alternatively, maybe I can compute the area by using the formula for equiangular polygons.

In general, for an equiangular polygon, even if it's not regular, there is a formula involving the side lengths and the angles. However, I don't remember it exactly.

Alternatively, since all angles are 135°, which is 180° - 45°, maybe we can model the octagon as a combination of vectors with angles 45° apart.

Alternatively, think of the octagon as a rectangle with side lengths a and b, with four isosceles right triangles cut off from each corner. Wait, but in a regular octagon, that's the case. However, in this case, the sides are not equal.

Wait, in a regular octagon, if you have a square and cut off the corners, each corner cut is an isosceles right triangle. However, in this case, the sides are of different lengths.

But maybe we can think of this octagon as a rectangle with some modifications.

Looking at the coordinates, the octagon has vertices at (0,0), (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), (0, 2), (-1/2, 3/2), (-1/2, 1/2), back to (0,0).

If I imagine this, the figure seems to have a sort of symmetry. Let me try to compute the area by breaking it into parts.

First, consider the rectangle from (-1/2, 1/2) to (3/2, 3/2). Wait, but that might not be helpful.

Alternatively, divide the octagon into the following parts:

1. The rectangle from (0,0) to (1,0) to (1,2) to (0,2) to (0,0). Wait, but that rectangle would have area 1*2 = 2. However, the octagon has more area.

But the octagon also includes the areas from (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) and similarly on the left side.

Alternatively, think of the octagon as a combination of a central rectangle and four trapezoids.

Alternatively, since I have coordinates, maybe use the shoelace formula again.

Alternatively, maybe I can check with vectors.

Alternatively, think of the octagon as a combination of squares and triangles.

But given that the shoelace formula gives 7/2, and I verified the coordinates, maybe that is correct.

But let me check with another coordinate system.

Alternatively, maybe I can compute the area by using the formula for the area of a polygon with sides and angles.

In general, for a polygon with sides a1, a2,..., an and angles θ1, θ2,..., θn, there is a formula, but I don't recall it.

Alternatively, since it's equiangular, all angles are equal, so maybe we can use some formula.

Alternatively, think of the octagon as a combination of vectors.

But since I already have the coordinates and the shoelace formula gives 7/2, which is 3.5, and the problem is from an Olympiad, maybe the answer is 7/2?

But let me check with another approach.

Let me compute the area by dividing the octagon into triangles.

For example, pick a point inside the octagon and connect it to all vertices. However, without knowing the exact position, it might be complex.

Alternatively, since we have coordinates, maybe divide the octagon into simpler polygons.

Looking at the coordinates:

From (0,0) to (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2) to (0, 2) to (-1/2, 3/2) to (-1/2, 1/2) to (0,0).

Let me divide this into two trapezoids and a rectangle?

Alternatively, divide it into the following parts:

1. The rectangle from (0,0) to (1,0) to (1,2) to (0,2) to (0,0). Area = 1*2 = 2.

2. The area from (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2). This is a trapezoid? Or maybe a parallelogram?

Wait, the points (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2).

This shape is a parallelogram? Let me check. The vector from (1,0) to (3/2, 1/2) is (1/2, 1/2). The vector from (1,0) to (1, 2) is (0, 2). But the other sides: from (3/2, 1/2) to (3/2, 3/2) is (0, 1), and from (3/2, 3/2) to (1, 2) is (-1/2, 1/2).

Alternatively, maybe compute the area of this quadrilateral.

Using shoelace formula for this quadrilateral:

Coordinates: (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), back to (1,0).

Compute terms:

Term1: 1*(1/2) - 3/2*0 = 1/2 - 0 = 1/2

Term2: 3/2*(3/2) - 3/2*(1/2) = 9/4 - 3/4 = 6/4 = 3/2

Term3: 3/2*2 - 1*(3/2) = 3 - 3/2 = 3/2

Term4: 1*0 - 1*2 = 0 - 2 = -2

Sum: 1/2 + 3/2 + 3/2 - 2 = (1/2 + 3/2) + (3/2 - 2) = 2 + (-1/2) = 3/2. Then area is 1/2 * |3/2| = 3/4.

Wait, but that seems small. Wait, let me check again:

Wait, shoelace formula for quadrilateral:

List the coordinates:

(1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), (1,0)

Compute xi*yi+1 - xi+1*yi for each i:

i=0: x0=1, y0=0; x1=3/2, y1=1/2: 1*(1/2) - 3/2*0 = 1/2 - 0 = 1/2

i=1: x1=3/2, y1=1/2; x2=3/2, y2=3/2: 3/2*(3/2) - 3/2*(1/2) = 9/4 - 3/4 = 6/4 = 3/2

i=2: x2=3/2, y2=3/2; x3=1, y3=2: 3/2*2 - 1*(3/2) = 3 - 3/2 = 3/2

i=3: x3=1, y3=2; x4=1, y4=0: 1*0 - 1*2 = 0 - 2 = -2

Sum: 1/2 + 3/2 + 3/2 - 2 = (1/2 + 3/2 + 3/2) - 2 = ( (1 + 3 + 3)/2 ) - 2 = 7/2 - 2 = 7/2 - 4/2 = 3/2.

Then area is 1/2 * |3/2| = 3/4.

So the area of this quadrilateral is 3/4. Similarly, there is a symmetrical quadrilateral on the left side.

Similarly, the left side has points (-1/2, 3/2), (-1/2, 1/2), (0,0), (0, 2)? Wait, no. Wait, the left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2)? No, the octagon's left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0)? Wait, no.

Wait, the octagon's vertices are:

(0,0), (1,0), (3/2, 1/2), (3/2, 3/2), (1, 2), (0, 2), (-1/2, 3/2), (-1/2, 1/2), back to (0,0).

So the left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0). Wait, but that's a triangle?

Wait, maybe I need to consider symmetry. The area on the right side (the quadrilateral we calculated) is 3/4, and there is a symmetrical area on the left side.

Similarly, there are two more areas on the top and bottom?

Alternatively, maybe the entire area is the rectangle (area 2) plus two quadrilaterals (each 3/4) plus two triangles?

Wait, let me think again.

The octagon can be divided into:

1. The central rectangle from (0,0) to (1,0) to (1,2) to (0,2) to (0,0). Area = 2.

2. The two trapezoids on the right and left sides.

Wait, the right trapezoid is from (1,0) to (3/2, 1/2) to (3/2, 3/2) to (1, 2). Which we calculated as 3/4. Similarly, the left trapezoid would be from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2)? Wait, no.

Wait, the left side has points (-1/2, 3/2), (-1/2, 1/2), (0,0). But how does that connect?

Wait, the octagon's left side is from (-1/2, 3/2) to (-1/2, 1/2) to (0,0). So that's a triangle?

Wait, if I consider the left side, from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2) to (1, 2) to (1, 0)...

Alternatively, maybe the left side has a similar area to the right side.

Let me compute the area of the left trapezoid.

The left trapezoid would be from (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2) to (-1/2, 3/2). Wait, but that's not a trapezoid.

Alternatively, maybe the area on the left is a triangle and a rectangle?

Alternatively, maybe I need to compute the area of the left side.

Let me take the points (-1/2, 3/2), (-1/2, 1/2), (0,0), and (0, 2).

Wait, if I connect (-1/2, 3/2) to (-1/2, 1/2) to (0,0) to (0, 2) to (-1/2, 3/2).

This is a quadrilateral. Let me apply the shoelace formula to this quadrilateral.

Coordinates: (-1/2, 3/2), (-1/2, 1/2), (0, 0), (0, 2), back to (-1/2, 3/2).

Compute terms:

Term1: (-1/2)*(1/2) - (-1/2)*(3/2) = (-1/4) - (-3/4) = (-1/4 + 3/4) = 2/4 = 1/2

Term2: (-1/2)*0 - 0*(1/2) = 0 - 0 = 0

Term3: 0*2 - 0*0 = 0 - 0 = 0

Term4: 0*(3/2) - (-1/2)*2 = 0 - (-1) = 1

Sum: 1/2 + 0 + 0 + 1 = 3/2. Area is 1/2 * |3/2| = 3/4.

So the left side also has an area of 3/4.

Therefore, total area would be central rectangle (2) + right trapezoid (3/4) + left trapezoid (3/4) = 2 + 3/4 + 3/4 = 2 + 1.5 = 3.5 = 7/2.

Which matches the shoelace formula result.

Therefore, the area is indeed 7/2.

But let me check once more.

Alternatively, think of the entire figure as a combination of the central rectangle (area 2) and two trapezoids (each 3/4), totaling 2 + 3/4 + 3/4 = 3.5.

Alternatively, maybe there are more areas?

Wait, the octagon also has the top and bottom parts?

Wait, no, because the central rectangle already includes the area from y=0 to y=2. The trapezoids on the sides are the extensions on the right and left.

Alternatively, maybe there are also areas on the top and bottom?

Wait, looking at the coordinates, the octagon's topmost point is at y=2, and the bottommost is at y=0. So the central rectangle already covers from y=0 to y=2.

Therefore, the total area is indeed 7/2.

Therefore, I think the answer is 7/2.

But let me check with another method.

Alternatively, since the octagon can be considered as a regular octagon with some sides adjusted.

But I don't know the side lengths of a regular octagon. However, maybe I can compute the area of the regular octagon and adjust it?

But since the given octagon is not regular, this might not be straightforward.

Alternatively, think of the octagon as a combination of squares and rectangles.

But given that I have two different methods giving me 7/2, I think that is correct.

Therefore, the area of the octagon is 7/2, which is 3.5.

But let me check if the problem says "four sides of length 1 and four sides of length √2/2". In my calculation, the sides are alternating between 1 and √2/2, which matches the problem's condition.

Therefore, I think the answer is 7/2.

But just to be thorough, let me check if there's any possible mistake in the shoelace formula.

The shoelace formula is reliable if the coordinates are correct. Since I walked through the coordinates step by step and they close properly, and the shoelace formula gives 7/2, which is confirmed by breaking into parts, I think it's safe.

Therefore, the area of the octagon is 7/2.

**Final Answer**
\boxed{\dfrac{7}{2}}

To find the area of the given equiangular octagon, we begin by understanding its structure and properties.

---

### **Understanding the Octagon**

An **equiangular octagon** has all internal angles equal. For an octagon, each internal angle is:

$$
\text{Internal angle} = \frac{(8 - 2) \cdot 180^\circ}{8} = 135^\circ
$$

This means that the **external angle** at each vertex is:

$$
\text{External angle} = 180^\circ - 135^\circ = 45^\circ
$$

Since the octagon is equiangular, the direction of each side changes by 45° from the previous one. The side lengths alternate between 1 and $\frac{\sqrt{2}}{2}$, and no two consecutive sides have the same length. This implies the sides alternate in the order:

$$
1, \frac{\sqrt{2}}{2}, 1, \frac{\sqrt{2}}{2}, \dots
$$

---

### **Modeling the Octagon as Vectors**

We can model the octagon as a closed polygon in the complex plane, where each side is a vector with a specific length and direction. Starting from the origin, we can compute the coordinates of each vertex by summing the vectors.

Let’s define the direction of each side as follows:

- Side 1: length 1, angle 0°
- Side 2: length $\frac{\sqrt{2}}{2}$, angle 45°
- Side 3: length 1, angle 90°
- Side 4: length $\frac{\sqrt{2}}{2}$, angle 135°
- Side 5: length 1, angle 180°
- Side 6: length $\frac{\sqrt{2}}{2}$, angle 225°
- Side 7: length 1, angle 270°
- Side 8: length $\frac{\sqrt{2}}{2}$, angle 315°

We convert each side into a vector in the complex plane and sum them up. The result is zero, confirming that the octagon is closed.

---

### **Computing the Area Using the Shoelace Formula**

We now compute the coordinates of the vertices by summing the vectors step by step. The resulting coordinates are:

- $V_0 = (0, 0)$
- $V_1 = (1, 0)$
- $V_2 = \left(\frac{3}{2}, \frac{1}{2}\right)$
- $V_3 = \left(\frac{3}{2}, \frac{3}{2}\right)$
- $V_4 = (1, 2)$
- $V_5 = (0, 2)$
- $V_6 = \left(-\frac{1}{2}, \frac{3}{2}\right)$
- $V_7 = \left(-\frac{1}{2}, \frac{1}{2}\right)$
- $V_8 = (0, 0)$

Using the **shoelace formula**:

$$
\text{Area} = \frac{1}{2} \left| \sum_{i=0}^{7} (x_i y_{i+1} - x_{i+1} y_i) \right|
$$

We compute each term and sum them up:

$$
\text{Sum} = 0 + \frac{1}{2} + \frac{3}{2} + \frac{3}{2} + 2 + 1 + \frac{1}{2} + 0 = 7
$$

$$
\text{Area} = \frac{1}{2} \cdot 7 = \frac{7}{2}
$$

---

### **Conclusion**

After verifying the coordinates and applying the shoelace formula, we find that the area of the equiangular octagon with alternating side lengths of 1 and $\frac{\sqrt{2}}{2}$ is:

$$
\boxed{\dfrac{7}{2}}
$$

Solution:
The octagon can be partitioned into five squares and four half squares, each with side length $\sqrt{2}/2$, so its area is \[
\displaystyle\left(5+4 \cdot \frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\displaystyle\right)^{2}= \boxed{\frac{7}{2}}.
\][asy]
unitsize(2cm);
for (int i=0; i<4; ++i) {
for (int j=0; j<4; ++j) {
draw((i,0)--(i,3),dashed);
draw((0,j)--(3,j),dashed);
};}
draw((1,0)--(2,0)--(3,1)--(3,2)--(2,3)--(1,3)--(0,2)--(0,1)--cycle,linewidth(0.7));
for (int i=0; i<2; ++i) {
label("1",(0.5+2i,2.5),S);
label("1",(0.5+2i,0.5),N);}
label("$\frac{\sqrt{2}}{2}$",(0,1.5),E);
label("$\frac{\sqrt{2}}{2}$",(3,1.5),W);
[/asy]


Explanation:
Okay, so I need to solve this problem where there's a sequence defined by the recurrence relation a_{i+1} = 1 / (1 - a_i) for i ≥ 1. And it's given that a_3 = a_1. I need to compute (a_9)^9. Hmm, let me think about how to approach this.

First, let me write down what I know. The sequence is defined recursively, so each term depends on the previous one. The relation is a_{i+1} = 1 / (1 - a_i). Also, we know that the third term is equal to the first term. So, a_3 = a_1. My goal is to find (a_9)^9.

Since the sequence is defined recursively, maybe I can express a_3 in terms of a_1, set it equal to a_1, and solve for a_1? Then once I have a_1, I can find a_9? Let me try that.

Let me start by writing expressions for a_2, a_3, etc., in terms of a_1.

Given a_1, then:

a_2 = 1 / (1 - a_1)

Then, a_3 = 1 / (1 - a_2) = 1 / [1 - (1 / (1 - a_1))]

Let me compute that denominator first. Let me compute 1 - a_2:

1 - a_2 = 1 - [1 / (1 - a_1)] = [ (1 - a_1) - 1 ] / (1 - a_1) ) = [ -a_1 ] / (1 - a_1 )

Therefore, a_3 = 1 / [ -a_1 / (1 - a_1) ) ] = [ (1 - a_1) / (-a_1) ) ] = (a_1 - 1)/a_1

So, a_3 = (a_1 - 1)/a_1

But we are told that a_3 = a_1. Therefore:

(a_1 - 1)/a_1 = a_1

Multiply both sides by a_1:

a_1 - 1 = a_1^2

Rearranged:

a_1^2 - a_1 + 1 = 0

Wait, hold on. Let me check that again. If (a_1 - 1)/a_1 = a_1, then multiplying both sides by a_1 gives:

a_1 - 1 = a_1^2

Which rearranged is:

a_1^2 - a_1 + 1 = 0? Wait, no. If we have a_1 - 1 = a_1^2, then bringing all terms to one side:

a_1^2 - a_1 + 1 = 0? Wait, no. Let me do that again.

Original equation after multiplying both sides by a_1:

Left side: a_1 - 1

Right side: a_1^2

Therefore, bringing all terms to left side:

a_1 - 1 - a_1^2 = 0 => -a_1^2 + a_1 - 1 = 0 => multiplying both sides by -1:

a_1^2 - a_1 + 1 = 0

Yes, that's correct. So quadratic equation: a_1^2 - a_1 + 1 = 0.

Wait, but solving this quadratic equation, discriminant D = (-1)^2 - 4*1*1 = 1 - 4 = -3 < 0. So the solutions are complex numbers. But the problem didn't specify whether the sequence is real or complex. Hmm. Wait, maybe I made a mistake in my calculation?

Let me check again step by step.

Starting with a_1.

a_2 = 1 / (1 - a_1)

Then a_3 = 1 / (1 - a_2) = 1 / [1 - 1/(1 - a_1)]

Let me compute denominator:

1 - [1 / (1 - a_1)] = [ (1 - a_1) - 1 ] / (1 - a_1) ) = (-a_1) / (1 - a_1)

Therefore, a_3 = 1 / [ (-a_1)/(1 - a_1) ) ] = (1 - a_1)/(-a_1) = (a_1 - 1)/a_1. Yes, that's correct.

So, a_3 = (a_1 - 1)/a_1. Then setting equal to a_1:

(a_1 - 1)/a_1 = a_1 => a_1 - 1 = a_1^2 => a_1^2 - a_1 + 1 = 0. So that's correct.

So the quadratic equation is a_1^2 - a_1 + 1 = 0. Which has complex roots. So maybe the problem is expecting complex numbers? Or perhaps I made a mistake in interpreting the problem?

Wait, the problem says "a sequence (a_n)" but doesn't specify if it's real or complex. Maybe it's okay. Let me check if there's another way. Maybe I made a mistake in the recurrence?

Wait, let me check again. Let me compute a_3 again. Maybe I miscalculated.

Given a_1, then a_2 = 1/(1 - a_1). Then a_3 = 1/(1 - a_2) = 1 / [1 - 1/(1 - a_1)].

Let me compute 1 - 1/(1 - a_1):

Let me write 1 as (1 - a_1)/(1 - a_1), so:

[ (1 - a_1) - 1 ] / (1 - a_1) ) = (-a_1)/(1 - a_1). So that's correct. Therefore, a_3 = 1 divided by that, which is (1 - a_1)/(-a_1) = (a_1 - 1)/a_1. Correct.

So, then setting equal to a_1:

(a_1 - 1)/a_1 = a_1 => a_1 - 1 = a_1^2 => a_1^2 - a_1 + 1 = 0. So that's correct.

So, the solutions are [1 ± sqrt(-3)] / 2, which is [1 ± i√3]/2. So complex numbers. So maybe the sequence is complex? But then, the problem says "compute (a_9)^9". If a_9 is complex, then raising it to the 9th power... but maybe there's some periodicity?

Alternatively, maybe there's a mistake in my calculation? Let me check again.

Alternatively, maybe the problem is in real numbers, and there's a mistake in my calculation? Let me check if there's another way to get a_3 = a_1.

Alternatively, maybe the sequence is periodic with period 2? Let me check.

Suppose that a_3 = a_1. Then, if the sequence is periodic with period 2, then a_1 = a_3 = a_5 = a_7 = a_9, so a_9 = a_1, so (a_9)^9 = (a_1)^9. But then, if I can find a_1, then compute (a_1)^9. But since a_1 is complex, maybe there's a way to compute that?

Alternatively, maybe there's a cycle of longer period? Let me check.

Alternatively, maybe I can find the period of the sequence. Let me try to compute terms in terms of a_1.

Let me compute a_1, a_2, a_3, a_4, etc., in terms of a_1.

We have:

a_1

a_2 = 1 / (1 - a_1)

a_3 = 1 / (1 - a_2) = (a_1 - 1)/a_1

Then a_4 = 1 / (1 - a_3) = 1 / [1 - (a_1 - 1)/a_1] = 1 / [ (a_1 - (a_1 - 1))/a_1 ] = 1 / [ (1)/a_1 ] = a_1

Wait, so a_4 = a_1? So if a_4 = a_1, then the sequence is periodic with period 3? Wait, because a_1, a_2, a_3, a_1, a_2, a_3,...

But given that a_3 = a_1, then a_4 = 1 / (1 - a_3) = 1 / (1 - a_1) = a_2. Then a_5 = 1 / (1 - a_4) = 1 / (1 - a_2) = a_3 = a_1. So yes, the sequence is periodic with period 3. Therefore, the sequence cycles every 3 terms. Therefore, a_1 = a_4 = a_7 = a_{10}, etc. Therefore, a_9 would be... Let's see:

Since the period is 3, then term number n: a_n = a_{n mod 3}. Wait, but need to check.

Wait, let's index:

n: 1, 2, 3, 4, 5, 6, 7, 8, 9,...

a_n: a1, a2, a3, a1, a2, a3, a1, a2, a3,...

So, for n=1: 1 mod 3 = 1, n=2: 2 mod 3 = 2, n=3: 0 mod 3? Wait, depends on how you define it. Let me think.

If the period is 3, then a_{k} = a_{k + 3} for all k ≥ 1. Therefore, a_9 = a_{9 - 3*3} = a_0? Wait, but the sequence starts at n=1. So maybe better to think:

Since a_1 = a_4 = a_7 = a_{10} = ..., so for n ≡ 1 mod 3, a_n = a_1.

Similarly, n ≡ 2 mod 3: a_n = a_2.

n ≡ 0 mod 3: a_n = a_3.

Therefore, since 9 divided by 3 is 3, so 9 ≡ 0 mod 3. Therefore, a_9 = a_3 = a_1. Therefore, (a_9)^9 = (a_1)^9.

But since a_3 = a_1, and we have from earlier that a_1^2 - a_1 + 1 = 0. Therefore, we can find a_1^9?

But since a_1 is a complex number, maybe we can find a_1^9 using the relation a_1^2 = a_1 - 1.

Let me try that. Let me note that from the equation a_1^2 = a_1 - 1. So, we can use this to find higher powers of a_1.

So, let me compute a_1^2 = a_1 - 1.

Then, a_1^3 = a_1 * a_1^2 = a_1*(a_1 - 1) = a_1^2 - a_1 = (a_1 - 1) - a_1 = -1.

Oh! So a_1^3 = -1. Therefore, a_1^3 = -1. Therefore, a_1^9 = (a_1^3)^3 = (-1)^3 = -1. Therefore, (a_9)^9 = (a_1)^9 = -1. Therefore, the answer is -1?

Wait, that seems straightforward. Let me check again.

Given that a_1^2 = a_1 - 1.

Then, a_1^3 = a_1 * a_1^2 = a_1*(a_1 - 1) = a_1^2 - a_1 = (a_1 - 1) - a_1 = -1. Yes, that's correct. Therefore, a_1^3 = -1. Therefore, a_1^9 = (a_1^3)^3 = (-1)^3 = -1. Therefore, (a_9)^9 = -1. Therefore, the answer is \boxed{-1}?

But let me check again. Since a_9 = a_3 = a_1, so (a_9)^9 = (a_1)^9 = -1. So that's the answer?

But let me check if there's another way to get this. Let me check with the roots of the equation a_1^2 - a_1 + 1 = 0. The roots are [1 ± i√3]/2. Let me compute a_1^3 for one of them.

Let me take ω = [1 + i√3]/2. This is a complex number. Let me compute ω^3.

Note that ω is a primitive 6th root of unity. Because ω^3 = e^(iπ/3)^3 = e^(iπ) = -1. Wait, actually, ω is e^(iπ/3), since cos(π/3) = 1/2, sin(π/3) = √3/2. Therefore, ω^3 = e^(iπ) = -1. Therefore, ω^3 = -1. Similarly, the other root is ω' = [1 - i√3]/2, which is e^(-iπ/3), and similarly (ω')^3 = -1. Therefore, indeed, a_1^3 = -1. Therefore, a_1^9 = (-1)^3 = -1. Therefore, regardless of which root we take, (a_1)^9 = -1. Therefore, the answer is -1. Therefore, (a_9)^9 = -1. Therefore, boxed answer is \boxed{-1}.

But let me check again if the sequence is indeed periodic with period 3. Let me take an example. Suppose a_1 is ω, then a_2 = 1/(1 - ω). Let me compute that.

Let me compute 1 - ω. Since ω = [1 + i√3]/2, then 1 - ω = 1 - [1 + i√3]/2 = [2 - 1 - i√3]/2 = [1 - i√3]/2. Which is ω'. Therefore, a_2 = 1 / (1 - ω) = 1 / ω' = ω''? Wait, since ω' is [1 - i√3]/2, which is e^{-iπ/3}, so 1/ω' is e^{iπ/3} = ω. Wait, no:

Wait, ω' is e^{-iπ/3}, so 1/ω' is e^{iπ/3} which is ω. Therefore, a_2 = ω. Wait, but that would mean a_2 = ω, which is a_1? But that can't be. Wait, no:

Wait, let me compute 1 / (1 - ω). Let me compute 1 - ω = [1 - (1 + i√3)/2] = [ (2 - 1 - i√3)/2 ] = [1 - i√3]/2. So 1 / (1 - ω) = 2 / (1 - i√3). Multiply numerator and denominator by (1 + i√3):

[2*(1 + i√3)] / [ (1)^2 - (i√3)^2 ] = [2 + 2i√3] / [1 - (-3)] = [2 + 2i√3]/4 = [1 + i√3]/2 = ω. Therefore, a_2 = ω. But that would mean a_2 = a_1? But that contradicts the previous calculation where a_2 = 1/(1 - a_1). Wait, but if a_1 = ω, then a_2 = 1/(1 - ω) = ω. Therefore, a_2 = a_1? But then a_3 = 1/(1 - a_2) = 1/(1 - ω) = ω = a_1. Therefore, the sequence would be a_1, a_1, a_1,...? But that contradicts the previous calculation where a_3 = (a_1 - 1)/a_1. Wait, but if a_1^2 - a_1 + 1 = 0, then (a_1 - 1)/a_1 = a_1. So that's consistent. But if a_2 = a_1, then a_3 = 1/(1 - a_2) = 1/(1 - a_1) = a_2 = a_1. Therefore, the sequence is constant? But that would mean that a_1 = a_2 = a_3 = ... But if that's the case, then the recurrence relation would require that a_1 = 1/(1 - a_1). Let me check that. If a_1 = 1/(1 - a_1), then multiplying both sides by (1 - a_1):

a_1(1 - a_1) = 1 => a_1 - a_1^2 = 1 => -a_1^2 + a_1 - 1 = 0 => a_1^2 - a_1 + 1 = 0. Which is the same equation as before. Therefore, if a_1 satisfies this equation, then the sequence is constant? Wait, but earlier when I calculated a_2, I thought that a_2 = 1/(1 - a_1) = ω, which is equal to a_1. Therefore, the sequence is constant? But then, if the sequence is constant, then a_3 = a_1, which is given. So that's consistent. But earlier, when I thought that the sequence is periodic with period 3, that was a mistake? Wait, but if the sequence is constant, then all terms are equal. But according to the previous calculation, when I calculated a_4, I thought that a_4 = a_1. But if the sequence is constant, then a_4 = a_1. So that's consistent. So maybe the sequence is constant? But how?

Wait, if a_1 = a_2 = a_3 = ..., then the recurrence relation is satisfied because a_{i+1} = 1/(1 - a_i) = a_i. Therefore, 1/(1 - a_i) = a_i => same equation. Therefore, the sequence is constant. Therefore, in this case, a_9 = a_1, so (a_9)^9 = (a_1)^9 = -1. Therefore, the answer is -1. So that seems to check out.

But then why did I think earlier that the sequence is periodic with period 3? Because when I calculated a_4, I thought that a_4 = a_1. But if the sequence is constant, then a_4 = a_1. So that's consistent. So maybe the sequence is constant? But then why does the recurrence relation lead to a_3 = a_1? Because if the sequence is constant, then a_3 = a_1 is automatically true. Therefore, the condition a_3 = a_1 is just saying that the sequence is constant. Therefore, the problem reduces to finding a_1 such that a_1 = 1/(1 - a_1), which gives the quadratic equation, and then computing (a_1)^9. Which we found to be -1. Therefore, the answer is -1.

But let me check with actual numbers. Let me take a_1 = [1 + i√3]/2. Then compute a_2 = 1/(1 - a_1). As above, that equals a_1. Therefore, a_2 = a_1. Therefore, the sequence is constant. Therefore, a_3 = a_1, which is given. Therefore, all terms are equal to a_1. Therefore, a_9 = a_1, so (a_9)^9 = (a_1)^9 = -1. Therefore, answer is -1.

Alternatively, if I take a_1 = [1 - i√3]/2, same thing. Therefore, regardless of which root we take, the result is the same. Therefore, the answer is -1.

But let me check with another approach. Since the sequence is constant, then a_1 = 1/(1 - a_1). Therefore, solving for a_1:

Multiply both sides by (1 - a_1):

a_1(1 - a_1) = 1 => a_1 - a_1^2 = 1 => -a_1^2 + a_1 - 1 = 0 => a_1^2 - a_1 + 1 = 0. Which is the same equation. Therefore, the roots are complex numbers as before. Therefore, the answer is -1.

Alternatively, maybe there's a mistake in assuming that the sequence is constant? Let me check with another example. Suppose that a_1 is some value, not necessarily satisfying the equation. For example, take a_1 = 0. Then a_2 = 1/(1 - 0) = 1. Then a_3 = 1/(1 - 1) which is undefined. So that's not allowed. If a_1 = 2, then a_2 = 1/(1 - 2) = -1. Then a_3 = 1/(1 - (-1)) = 1/2. Then a_4 = 1/(1 - 1/2) = 2. Then a_5 = 1/(1 - 2) = -1, etc. So here, the sequence is periodic with period 3: 2, -1, 1/2, 2, -1, 1/2,... So in this case, a_3 = 1/2 ≠ a_1 = 2. Therefore, in this case, the condition a_3 = a_1 would not hold. Therefore, only when a_1 satisfies the equation a_1^2 - a_1 + 1 = 0, the sequence becomes constant. Therefore, in that case, the sequence is constant, so all terms are equal to a_1, so a_9 = a_1, and (a_9)^9 = (a_1)^9 = -1. Therefore, answer is -1.

Therefore, I think the answer is \boxed{-1}.

**Final Answer**
\boxed{-1}

To solve the problem, we are given a recursive sequence defined by:

$$
a_{i+1} = \frac{1}{1 - a_i}, \quad \text{for } i \geq 1
$$

and the condition that $ a_3 = a_1 $. We are asked to compute $ (a_9)^9 $.

---

### Step 1: Express $ a_3 $ in terms of $ a_1 $

We compute the first few terms of the sequence in terms of $ a_1 $:

- $ a_2 = \frac{1}{1 - a_1} $
- $ a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - \frac{1}{1 - a_1}} $

Simplify the denominator:

$$
1 - \frac{1}{1 - a_1} = \frac{(1 - a_1) - 1}{1 - a_1} = \frac{-a_1}{1 - a_1}
$$

So:

$$
a_3 = \frac{1}{\frac{-a_1}{1 - a_1}} = \frac{1 - a_1}{-a_1} = \frac{a_1 - 1}{a_1}
$$

Given that $ a_3 = a_1 $, we equate:

$$
\frac{a_1 - 1}{a_1} = a_1
$$

Multiply both sides by $ a_1 $:

$$
a_1 - 1 = a_1^2
$$

Rearranging:

$$
a_1^2 - a_1 + 1 = 0
$$

---

### Step 2: Solve the quadratic equation

The quadratic equation is:

$$
a_1^2 - a_1 + 1 = 0
$$

Using the quadratic formula:

$$
a_1 = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(1)}}{2} = \frac{1 \pm \sqrt{-3}}{2}
$$

So the roots are:

$$
a_1 = \frac{1 \pm i\sqrt{3}}{2}
$$

These are complex numbers, specifically the primitive sixth roots of unity. Let’s denote:

$$
\omega = \frac{1 + i\sqrt{3}}{2}, \quad \omega' = \frac{1 - i\sqrt{3}}{2}
$$

Note that $ \omega^3 = -1 $, and similarly $ (\omega')^3 = -1 $.

---

### Step 3: Determine the value of $ a_9 $

From the recurrence relation, we can observe the behavior of the sequence. Let’s compute the first few terms:

- $ a_1 = \omega $
- $ a_2 = \frac{1}{1 - \omega} = \omega $
- $ a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - \omega} = \omega $

So the sequence becomes constant:

$$
a_1 = a_2 = a_3 = a_4 = \cdots = \omega
$$

Therefore, $ a_9 = a_1 = \omega $

---

### Step 4: Compute $ (a_9)^9 $

Since $ a_9 = a_1 = \omega $, we compute:

$$
(a_9)^9 = \omega^9
$$

But we know that $ \omega^3 = -1 $, so:

$$
\omega^9 = (\omega^3)^3 = (-1)^3 = -1
$$

---

### Final Answer

$$
\boxed{-1}
$$

[Assistant]
Explanation:
Okay, so I need to solve this problem where there's a sequence defined by the recurrence relation a_{i+1} = 1 / (1 - a_i) for i ≥ 1. And it's given that a_3 = a_1. I need to compute (a_9)^9. Hmm, let me think about how to approach this.

First, let me write down what I know. The sequence is defined recursively, so each term depends on the previous one. The relation is a_{i+1} = 1 / (1 - a_i). Also, we know that the third term is equal to the first term. So, a_3 = a_1. My goal is to find (a_9)^9.

Since the sequence is defined recursively, maybe I can express a_3 in terms of a_1, set it equal to a_1, and solve for a_1? Then once I have a_1, I can find a_9? Let me try that.

Let me start by writing expressions for a_2, a_3, etc., in terms of a_1.

Given a_1, then:

a_2 = 1 / (1 - a_1)

Then, a_3 = 1 / (1 - a_2) = 1 / [1 - (1 / (1 - a_1))]

Let me compute that denominator first. Let me compute 1 - a_2:

1 - a_2 = 1 - [1 / (1 - a_1)] = [ (1 - a_1) - 1 ] / (1 - a_1) ) = [ -a_1 ] / (1 - a_1 )

Therefore, a_3 = 1 / [ -a_1 / (1 - a_1) ) ] = [ (1 - a_1) / (-a_1) ) ] = (a_1 - 1)/a_1

So, a_3 = (a_1 - 1)/a_1

But we are told that a_3 = a_1. Therefore:

(a_1 - 1)/a_1 = a_1

Multiply both sides by a_1:

a_1 - 1 = a_1^2

Rearranged:

a_1^2 - a_1 + 1 = 0

Wait, hold on. Let me check that again. If (a_1 - 1)/a_1 = a_1, then multiplying both sides by a_1 gives:

a_1 - 1 = a_1^2

Which rearranged is:

a_1^2 - a_1 + 1 = 0? Wait, no. If we have a_1 - 1 = a_1^2, then bringing all terms to one side:

a_1^2 - a_1 + 1 = 0? Wait, no. Let me do that again.

Original equation after multiplying both sides by a_1:

Left side: a_1 - 1

Right side: a_1^2

Therefore, bringing all terms to left side:

a_1 - 1 - a_1^2 = 0 => -a_1^2 + a_1 - 1 = 0 => multiplying both sides by -1:

a_1^2 - a_1 + 1 = 0

Yes, that's correct. So quadratic equation: a_1^2 - a_1 + 1 = 0.

Wait, but solving this quadratic equation, discriminant D = (-1)^2 - 4*1*1 = 1 - 4 = -3 < 0. So the solutions are complex numbers. But the problem didn't specify whether the sequence is real or complex. Hmm. Wait, maybe I made a mistake in my calculation?

Let me check again step by step.

Starting with a_1.

a_2 = 1 / (1 - a_1)

Then a_3 = 1 / (1 - a_2) = 1 / [1 - 1/(1 - a_1)]

Let me compute denominator:

1 - [1 / (1 - a_1)] = [ (1 - a_1) - 1 ] / (1 - a_1) ) = (-a_1) / (1 - a_1)

Therefore, a_3 = 1 / [ (-a_1)/(1 - a_1) ) ] = (1 - a_1)/(-a_1) = (a_1 - 1)/a_1. Yes, that's correct.

So, a_3 = (a_1 - 1)/a_1. Then setting equal to a_1:

(a_1 - 1)/a_1 = a_1 => a_1 - 1 = a_1^2 => a_1^2 - a_1 + 1 = 0. So that's correct.

So the quadratic equation is a_1^2 - a_1 + 1 = 0. Which has complex roots. So maybe the problem is expecting complex numbers? Or perhaps I made a mistake in interpreting the problem?

Wait, the problem says "a sequence (a_n)" but doesn't specify if it's real or complex. Maybe it's okay. Let me check if there's another way. Maybe I made a mistake in the recurrence?

Wait, let me check again. Let me compute a_3 again. Maybe I miscalculated.

Given a_1, then a_2 = 1/(1 - a_1). Then a_3 = 1/(1 - a_2) = 1 / [1 - 1/(1 - a_1)].

Let me compute 1 - 1/(1 - a_1):

Let me write 1 as (1 - a_1)/(1 - a_1), so:

[ (1 - a_1) - 1 ] / (1 - a_1) ) = (-a_1)/(1 - a_1). So that's correct. Therefore, a_3 = 1 divided by that, which is (1 - a_1)/(-a_1) = (a_1 - 1)/a_1. Correct.

So, then setting equal to a_1:

(a_1 - 1)/a_1 = a_1 => a_1 - 1 = a_1^2 => a_1^2 - a_1 + 1 = 0. So that's correct.

So, the solutions are [1 ± sqrt(-3)] / 2, which is [1 ± i√3]/2. So complex numbers. So maybe the sequence is complex? But then, the problem says "compute (a_9)^9". If a_9 is complex, then raising it to the 9th power... but maybe there's some periodicity?

Alternatively, maybe there's a mistake in my calculation? Let me check again.

Alternatively, maybe the problem is in real numbers, and there's a mistake in my calculation? Let me check if there's another way to get a_3 = a_1.

Alternatively, maybe the sequence is periodic with period 2? Let me check.

Suppose that a_3 = a_1. Then, if the sequence is periodic with period 2, then a_1 = a_3 = a_5 = a_7 = a_9, so a_9 = a_1, so (a_9)^9 = (a_1)^9. But then, if I can find a_1, then compute (a_1)^9. But since a_1 is complex, maybe there's a way to compute that?

Alternatively, maybe there's a cycle of longer period? Let me check.

Alternatively, maybe I can find the period of the sequence. Let me try to compute terms in terms of a_1.

Let me compute a_1, a_2, a_3, a_4, etc., in terms of a_1.

We have:

a_1

a_2 = 1 / (1 - a_1)

a_3 = 1 / (1 - a_2) = (a_1 - 1)/a_1

Then a_4 = 1 / (1 - a_3) = 1 / [1 - (a_1 - 1)/a_1] = 1 / [ (a_1 - (a_1 - 1))/a_1 ] = 1 / [ (1)/a_1 ] = a_1

Wait, so a_4 = a_1? So if a_4 = a_1, then the sequence is periodic with period 3? Wait, because a_1, a_2, a_3, a_1, a_2, a_3,...

But given that a_3 = a_1, then a_4 = 1 / (1 - a_3) = 1 / (1 - a_1) = a_2. Then a_5 = 1 / (1 - a_4) = 1 / (1 - a_2) = a_3 = a_1. So yes, the sequence is periodic with period 3. Therefore, the sequence cycles every 3 terms. Therefore, a_1 = a_4 = a_7 = a_{10}, etc. Therefore, a_9 would be... Let's see:

Since the period is 3, then term number n: a_n = a_{n mod 3}. Wait, but need to check.

Wait, let's index:

n: 1, 2, 3, 4, 5, 6, 7, 8, 9,...

a_n: a1, a2, a3, a1, a2, a3, a1, a2, a3,...

So, for n=1: 1 mod 3 = 1, n=2: 2 mod 3 = 2, n=3: 0 mod 3? Wait, depends on how you define it. Let me think.

If the period is 3, then a_{k} = a_{k + 3} for all k ≥ 1. Therefore, a_9 = a_{9 - 3*3} = a_0? Wait, but the sequence starts at n=1. So maybe better to think:

Since a_1 = a_4 = a_7 = a_{10} = ..., so for n ≡ 1 mod 3, a_n = a_1.

Similarly, n ≡ 2 mod 3: a_n = a_2.

n ≡ 0 mod 3: a_n = a_3.

Therefore, since 9 divided by 3 is 3, so 9 ≡ 0 mod 3. Therefore, a_9 = a_3 = a_1. Therefore, (a_9)^9 = (a_1)^9.

But since a_3 = a_1, and we have from earlier that a_1^2 - a_1 + 1 = 0. Therefore, we can find a_1^9?

But since a_1 is a complex number, maybe we can find a_1^9 using the relation a_1^2 = a_1 - 1.

Let me try that. Let me note that from the equation a_1^2 = a_1 - 1. So, we can use this to find higher powers of a_1.

So, let me compute a_1^2 = a_1 - 1.

Then, a_1^3 = a_1 * a_1^2 = a_1*(a_1 - 1) = a_1^2 - a_1 = (a_1 - 1) - a_1 = -1.

Oh! So a_1^3 = -1. Therefore, a_1^3 = -1. Therefore, a_1^9 = (a_1^3)^3 = (-1)^3 = -1. Therefore, (a_9)^9 = (a_1)^9 = -1. Therefore, the answer is -1?

Wait, that seems straightforward. Let me check again.

Given that a_1^2 = a_1 - 1.

Then, a_1^3 = a_1 * a_1^2 = a_1*(a_1 - 1) = a_1^2 - a_1 = (a_1 - 1) - a_1 = -1. Yes, that's correct. Therefore, a_1^3 = -1. Therefore, a_1^9 = (a_1^3)^3 = (-1)^3 = -1. Therefore, (a_9)^9 = -1. Therefore, the answer is \boxed{-1}?

But let me check again. Since a_9 = a_3 = a_1, so (a_9)^9 = (a_1)^9 = -1. So that's the answer?

But let me check if there's another way to get this. Let me check with the roots of the equation a_1^2 - a_1 + 1 = 0. The roots are [1 ± i√3]/2. Let me compute a_1^3 for one of them.

Let me take ω = [1 + i√3]/2. This is a complex number. Let me compute ω^3.

Note that ω is a primitive 6th root of unity. Because ω^3 = e^(iπ/3)^3 = e^(iπ) = -1. Wait, actually, ω is e^(iπ/3), since cos(π/3) = 1/2, sin(π/3) = √3/2. Therefore, ω^3 = e^(iπ) = -1. Therefore, ω^3 = -1. Similarly, the other root is ω' = [1 - i√3]/2, which is e^(-iπ/3), and similarly (ω')^3 = -1. Therefore, indeed, a_1^3 = -1. Therefore, a_1^9 = (-1)^3 = -1. Therefore, regardless of which root we take, (a_1)^9 = -1. Therefore, the answer is -1. Therefore, (a_9)^9 = -1. Therefore, boxed answer is \boxed{-1}.

But let me check again if the sequence is indeed periodic with period 3. Let me take an example. Suppose a_1 is ω, then a_2 = 1/(1 - ω). Let me compute that.

Let me compute 1 - ω. Since ω = [1 + i√3]/2, then 1 - ω = 1 - [1 + i√3]/2 = [2 - 1 - i√3]/2 = [1 - i√3]/2. Which is ω'. Therefore, a_2 = 1 / (1 - ω) = 1 / ω' = ω''? Wait, since ω' is [1 - i√3]/2, which is e^{-iπ/3}, so 1/ω' is e^{iπ/3} = ω. Wait, no:

Wait, ω' is e^{-iπ/3}, so 1/ω' is e^{iπ/3} which is ω. Therefore, a_2 = ω. Wait, but that would mean a_2 = ω, which is a_1? But that can't be. Wait, no:

Wait, let me compute 1 / (1 - ω). Let me compute 1 - ω = [1 - (1 + i√3)/2] = [ (2 - 1 - i√3)/2 ] = [1 - i√3]/2. So 1 / (1 - ω) = 2 / (1 - i√3). Multiply numerator and denominator by (1 + i√3):

[2*(1 + i√3)] / [ (1)^2 - (i√3)^2 ] = [2 + 2i√3] / [1 - (-3)] = [2 + 2i√3]/4 = [1 + i√3]/2 = ω. Therefore, a_2 = ω. But that would mean a_2 = a_1? But that contradicts the previous calculation where a_2 = 1/(1 - a_1). Wait, but if a_1 = ω, then a_2 = 1/(1 - ω) = ω. Therefore, a_2 = a_1? But then a_3 = 1/(1 - a_2) = 1/(1 - ω) = ω = a_1. Therefore, the sequence would be a_1, a_1, a_1,...? But that contradicts the previous calculation where a_3 = (a_1 - 1)/a_1. Wait, but if a_1^2 - a_1 + 1 = 0, then (a_1 - 1)/a_1 = a_1. So that's consistent. But if a_2 = a_1, then a_3 = 1/(1 - a_2) = 1/(1 - a_1) = a_2 = a_1. Therefore, the sequence is constant? But that would mean that a_1 = a_2 = a_3 = ... But if that's the case, then the recurrence relation would require that a_1 = 1/(1 - a_1). Let me check that. If a_1 = 1/(1 - a_1), then multiplying both sides by (1 - a_1):

a_1(1 - a_1) = 1 => a_1 - a_1^2 = 1 => -a_1^2 + a_1 - 1 = 0 => a_1^2 - a_1 + 1 = 0. Which is the same equation as before. Therefore, if a_1 satisfies this equation, then the sequence is constant? Wait, but earlier when I calculated a_2, I thought that a_2 = 1/(1 - a_1) = ω, which is equal to a_1. Therefore, the sequence is constant? But then, if the sequence is constant, then a_3 = a_1, which is given. So that's consistent. But earlier, when I thought that the sequence is periodic with period 3, that was a mistake? Wait, but if the sequence is constant, then all terms are equal. But according to the previous calculation, when I calculated a_4, I thought that a_4 = a_1. But if the sequence is constant, then a_4 = a_1. So that's consistent. So maybe the sequence is constant? But how?

Wait, if a_1 = a_2 = a_3 = ..., then the recurrence relation is satisfied because a_{i+1} = 1/(1 - a_i) = a_i. Therefore, 1/(1 - a_i) = a_i => same equation. Therefore, the sequence is constant. Therefore, in this case, a_9 = a_1, so (a_9)^9 = (a_1)^9 = -1. Therefore, the answer is -1. So that seems to check out.

But then why did I think earlier that the sequence is periodic with period 3? Because when I calculated a_4, I thought that a_4 = a_1. But if the sequence is constant, then a_4 = a_1. So that's consistent. So maybe the sequence is constant? But then why does the recurrence relation lead to a_3 = a_1? Because if the sequence is constant, then a_3 = a_1 is automatically true. Therefore, the condition a_3 = a_1 is just saying that the sequence is constant. Therefore, the problem reduces to finding a_1 such that a_1 = 1/(1 - a_1), which gives the quadratic equation, and then computing (a_1)^9. Which we found to be -1. Therefore, the answer is -1.

But let me check with actual numbers. Let me take a_1 = [1 + i√3]/2. Then compute a_2 = 1/(1 - a_1). As above, that equals a_1. Therefore, a_2 = a_1. Therefore, the sequence is constant. Therefore, a_3 = a_1, which is given. Therefore, all terms are equal to a_1. Therefore, a_9 = a_1, so (a_9)^9 = (a_1)^9 = -1. Therefore, answer is -1.

Alternatively, if I take a_1 = [1 - i√3]/2, same thing. Therefore, regardless of which root we take, the result is the same. Therefore, the answer is -1.

But let me check with another approach. Since the sequence is constant, then a_1 = 1/(1 - a_1). Therefore, solving for a_1:

Multiply both sides by (1 - a_1):

a_1(1 - a_1) = 1 => a_1 - a_1^2 = 1 => -a_1^2 + a_1 - 1 = 0 => a_1^2 - a_1 + 1 = 0. Which is the same equation. Therefore, the roots are complex numbers as before. Therefore, the answer is -1.

Alternatively, maybe there's a mistake in assuming that the sequence is constant? Let me check with another example. Suppose that a_1 is some value, not necessarily satisfying the equation. For example, take a_1 = 0. Then a_2 = 1/(1 - 0) = 1. Then a_3 = 1/(1 - 1) which is undefined. So that's not allowed. If a_1 = 2, then a_2 = 1/(1 - 2) = -1. Then a_3 = 1/(1 - (-1)) = 1/2. Then a_4 = 1/(1 - 1/2) = 2. Then a_5 = 1/(1 - 2) = -1, etc. So here, the sequence is periodic with period 3: 2, -1, 1/2, 2, -1, 1/2,... So in this case, a_3 = 1/2 ≠ a_1 = 2. Therefore, in this case, the condition a_3 = a_1 would not hold. Therefore, only when a_1 satisfies the equation a_1^2 - a_1 + 1 = 0, the sequence becomes constant. Therefore, in that case, the sequence is constant, so all terms are equal to a_1, so a_9 = a_1, and (a_9)^9 = (a_1)^9 = -1. Therefore, answer is -1.

Therefore, I think the answer is \boxed{-1}.

**Final Answer**
\boxed{-1}

To solve the problem, we are given a recursive sequence defined by:

$$
a_{i+1} = \frac{1}{1 - a_i}, \quad \text{for } i \geq 1
$$

and the condition that $ a_3 = a_1 $. We are asked to compute $ (a_9)^9 $.

---

### Step 1: Express $ a_3 $ in terms of $ a_1 $

We compute the first few terms of the sequence in terms of $ a_1 $:

- $ a_2 = \frac{1}{1 - a_1} $
- $ a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - \frac{1}{1 - a_1}} $

Simplify the denominator:

$$
1 - \frac{1}{1 - a_1} = \frac{(1 - a_1) - 1}{1 - a_1} = \frac{-a_1}{1 - a_1}
$$

So:

$$
a_3 = \frac{1}{\frac{-a_1}{1 - a_1}} = \frac{1 - a_1}{-a_1} = \frac{a_1 - 1}{a_1}
$$

Given that $ a_3 = a_1 $, we equate:

$$
\frac{a_1 - 1}{a_1} = a_1
$$

Multiply both sides by $ a_1 $:

$$
a_1 - 1 = a_1^2
$$

Rearranging:

$$
a_1^2 - a_1 + 1 = 0
$$

---

### Step 2: Solve the quadratic equation

The quadratic equation is:

$$
a_1^2 - a_1 + 1 = 0
$$

Using the quadratic formula:

$$
a_1 = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(1)}}{2} = \frac{1 \pm \sqrt{-3}}{2}
$$

So the roots are:

$$
a_1 = \frac{1 \pm i\sqrt{3}}{2}
$$

These are complex numbers, specifically the primitive sixth roots of unity. Let’s denote:

$$
\omega = \frac{1 + i\sqrt{3}}{2}, \quad \omega' = \frac{1 - i\sqrt{3}}{2}
$$

Note that $ \omega^3 = -1 $, and similarly $ (\omega')^3 = -1 $.

---

### Step 3: Determine the value of $ a_9 $

From the recurrence relation, we can observe the behavior of the sequence. Let’s compute the first few terms:

- $ a_1 = \omega $
- $ a_2 = \frac{1}{1 - \omega} = \omega $
- $ a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - \omega} = \omega $

So the sequence becomes constant:

$$
a_1 = a_2 = a_3 = a_4 = \cdots = \omega
$$

Therefore, $ a_9 = a_1 = \omega $

---

### Step 4: Compute $ (a_9)^9 $

Since $ a_9 = a_1 = \omega $, we compute:

$$
(a_9)^9 = \omega^9
$$

But we know that $ \omega^3 = -1 $, so:

$$
\omega^9 = (\omega^3)^3 = (-1)^3 = -1
$$

---

### Final Answer

$$
\boxed{-1}
$$

Solution:
First, if $a_3 = a_1,$ then
\[a_1 = a_3 = a_5 = a_7 = a_9,\]so $(a_9)^9 = (a_1)^9.$

We have that
\begin{align*}
a_2 &= \frac{1}{1 - a_1}, \\
a_3 &= \frac{1}{1 - a_2} = \frac{1}{1 - \frac{1}{1 - a_1}} = \frac{1 - a_1}{1 - a_1 - 1} = \frac{1 - a_1}{-a_1}.
\end{align*}Then
\[\frac{1 - a_1}{-a_1} = a_1,\]so $1 - a_1 = -a_1^2.$  Then $a_1^2 - a_1 + 1 = 0.$  Multiplying both sides by $a_1 + 1,$ we get
\[(a_1 + 1)(a_1 ^2 - a_1 + 1) = 0,\]so $a_1^3 + 1 = 0.$  Then $a_1^3 = -1,$ so $a_1^9 = (-1)^3 = \boxed{-1}.$