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. Also, no two consecutive sides have the same length. Let me try to visualize this first. An equiangular octagon means all the internal angles are equal. For a regular octagon, all sides and angles are equal, but this one is not regular since the sides are different. However, since it's equiangular, the angles are still all the same, which I think are 135 degrees each because in a regular octagon each internal angle is 135 degrees.

So, first, maybe I should recall the formula for the area of a regular octagon. But since this one is not regular, I can't directly use that formula. Maybe I can think of it as a regular octagon that's been stretched or something? But I need to figure out how the sides are arranged. The problem says there are 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 pattern of sides alternates between 1 and √2/2? Or maybe some other pattern?

Wait, there are four of each. Since no two consecutive sides can be the same, the sides must alternate between the two lengths. So maybe the pattern is 1, √2/2, 1, √2/2, etc. So the sides alternate between 1 and √2/2. Since it's an octagon, there are eight sides. So starting with 1, then √2/2, then 1, then √2/2, and so on. That would give four of each, and no two consecutive sides are the same. So that seems like a possible arrangement.

Now, if that's the case, maybe I can model this octagon as a modified regular octagon. Maybe by cutting off the corners of a square or something? Wait, but equiangular octagons can be thought of as similar to regular octagons but with sides of different lengths. Alternatively, maybe I can use coordinates. Let me think.

Since the octagon is equiangular, all the internal angles are 135 degrees. If I can model the octagon on a coordinate system, maybe using vectors or something. Let me try to think about how to compute the area.

Alternatively, maybe I can divide the octagon into simpler shapes like triangles, rectangles, or trapezoids. But since it's equiangular, perhaps there's some symmetry?

Wait, another approach: For an equiangular polygon, even if it's not regular, the sides can be related through vectors. Because each turn between sides is the same. For an octagon, each external angle is 45 degrees, since the sum of external angles is 360 degrees, so 360/8 = 45 degrees. Wait, but internal angles are 135, so external angles are 45. So when you walk around the octagon, at each vertex you turn 45 degrees.

If I think of the octagon as being constructed by vectors in the plane, each subsequent side is a vector obtained by rotating the previous one by 45 degrees. Wait, but the lengths of the sides are different. So maybe I can model the sides as vectors with angles 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, right? But since it's a closed polygon, the sum of all the vectors (sides) should be zero.

Therefore, if I denote the sides as vectors, each rotated by 45 degrees from the previous one, with lengths alternating between 1 and √2/2. Let me try to formalize this.

Let me denote the sides as vectors. Let me suppose that the first side is along the positive x-axis, length 1. Then the next side is at angle 45°, length √2/2. Then the next side is at angle 90°, length 1, then 135°, length √2/2, 180°, length 1, 225°, length √2/2, 270°, length 1, 315°, length √2/2, and then finally the last side should bring us back to the starting point. Wait, but there are 8 sides, so let me check:

Starting at angle 0°, then each subsequent side is rotated by 45°. So the angles of each side would be 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and then the eighth side would be at 315°, and then adding the eighth side should complete the polygon.

But let me check if that works. Let me denote the sides as vectors:

Let me define each side as a vector. Let me denote the sides as follows:

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°

Then, if we add up all these vectors, they should sum to zero. Let me check that. If this is the case, then the polygon closes.

So, let's compute the sum of these vectors.

First, let me write each vector in terms of components:

s₁ = 1*(cos 0°, sin 0°) = (1, 0)

s₂ = (√2/2)*(cos 45°, sin 45°) = (√2/2)*(√2/2, √2/2) = ( (√2 * √2)/4 , same ) = ( (2)/4, 2/4 ) = (0.5, 0.5)

Similarly, s₃ = 1*(cos 90°, sin 90°) = (0, 1)

s₄ = (√2/2)*(cos 135°, sin 135°) = (√2/2)*(-√2/2, √2/2) = (- (2)/4, (2)/4 ) = (-0.5, 0.5)

s₅ = 1*(cos 180°, sin 180°) = (-1, 0)

s₆ = (√2/2)*(cos 225°, sin 225°) = (√2/2)*(-√2/2, -√2/2) = (- (2)/4, - (2)/4 ) = (-0.5, -0.5)

s₇ = 1*(cos 270°, sin 270°) = (0, -1)

s₈ = (√2/2)*(cos 315°, sin 315°) = (√2/2)*(√2/2, -√2/2) = ( (2)/4, - (2)/4 ) = (0.5, -0.5)

Now let me sum up all these components:

First, sum of x-components:

s₁x + s₂x + s₃x + s₄x + s₅x + s₆x + s₇x + s₈x

= 1 + 0.5 + 0 + (-0.5) + (-1) + (-0.5) + 0 + 0.5

Compute step by step:

Start with 1.

Add 0.5: 1.5

Add 0: 1.5

Add (-0.5): 1.0

Add (-1): 0.0

Add (-0.5): -0.5

Add 0: -0.5

Add 0.5: 0.0

So total x-component sum is 0.

Similarly, y-components:

s₁y + s₂y + s₃y + s₄y + s₅y + s₆y + s₇y + s₈y

= 0 + 0.5 + 1 + 0.5 + 0 + (-0.5) + (-1) + (-0.5)

Compute step by step:

Start with 0.

Add 0.5: 0.5

Add 1: 1.5

Add 0.5: 2.0

Add 0: 2.0

Add (-0.5): 1.5

Add (-1): 0.5

Add (-0.5): 0.0

So y-component sum is also 0.

Therefore, the vectors do sum to zero. Therefore, this arrangement of sides actually forms a closed octagon. So that must be the correct configuration.

Therefore, the octagon can be represented by these vectors. Now, to compute its area. How can I compute the area of such a polygon?

One method to compute the area of a polygon given its vertices is the shoelace formula. For that, I need coordinates of the vertices. 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). Then, each subsequent vertex is obtained by adding the respective side vector.

Let me denote the vertices as P₀, P₁, P₂, ..., P₈, where P₀ is the starting point, and P₈ coincides with P₀ since it's a closed polygon.

Starting at P₀ = (0, 0).

Then:

P₁ = P₀ + s₁ = (0 + 1, 0 + 0) = (1, 0)

P₂ = P₁ + s₂ = (1 + 0.5, 0 + 0.5) = (1.5, 0.5)

P₃ = P₂ + s₃ = (1.5 + 0, 0.5 + 1) = (1.5, 1.5)

P₄ = P₃ + s₄ = (1.5 + (-0.5), 1.5 + 0.5) = (1.0, 2.0)

P₅ = P₄ + s₅ = (1.0 + (-1), 2.0 + 0) = (0.0, 2.0)

P₆ = P₅ + s₆ = (0.0 + (-0.5), 2.0 + (-0.5)) = (-0.5, 1.5)

P₇ = P₆ + s₇ = (-0.5 + 0, 1.5 + (-1)) = (-0.5, 0.5)

P₈ = P₇ + s₈ = (-0.5 + 0.5, 0.5 + (-0.5)) = (0.0, 0.0)

Which brings us back to P₀. Perfect, that works.

So now, the coordinates of the vertices are:

P₀: (0, 0)

P₁: (1, 0)

P₂: (1.5, 0.5)

P₃: (1.5, 1.5)

P₄: (1.0, 2.0)

P₅: (0.0, 2.0)

P₆: (-0.5, 1.5)

P₇: (-0.5, 0.5)

P₈: (0, 0) (which is same as P₀)

Now, I can apply the shoelace formula to compute the area. The shoelace formula says that the area is half the absolute value of the sum over the vertices of (x_i y_{i+1} - x_{i+1} y_i).

Let me write down the coordinates again:

Let me list them in order:

P₀: (0, 0)

P₁: (1, 0)

P₂: (1.5, 0.5)

P₃: (1.5, 1.5)

P₄: (1.0, 2.0)

P₅: (0.0, 2.0)

P₆: (-0.5, 1.5)

P₇: (-0.5, 0.5)

P₈: (0, 0)

So, applying the shoelace formula:

Compute sum_{i=0}^7 (x_i y_{i+1} - x_{i+1} y_i)

Let me compute each term step by step.

Term 0: i=0, P₀ to P₁:

x0 y1 - x1 y0 = (0)(0) - (1)(0) = 0 - 0 = 0

Term 1: i=1, P₁ to P₂:

x1 y2 - x2 y1 = (1)(0.5) - (1.5)(0) = 0.5 - 0 = 0.5

Term 2: i=2, P₂ to P₃:

x2 y3 - x3 y2 = (1.5)(1.5) - (1.5)(0.5) = 2.25 - 0.75 = 1.5

Term 3: i=3, P₃ to P₄:

x3 y4 - x4 y3 = (1.5)(2.0) - (1.0)(1.5) = 3.0 - 1.5 = 1.5

Term 4: i=4, P₄ to P₅:

x4 y5 - x5 y4 = (1.0)(2.0) - (0.0)(2.0) = 2.0 - 0 = 2.0

Term 5: i=5, P₅ to P₆:

x5 y6 - x6 y5 = (0.0)(1.5) - (-0.5)(2.0) = 0 - (-1.0) = 1.0

Term 6: i=6, P₆ to P₇:

x6 y7 - x7 y6 = (-0.5)(0.5) - (-0.5)(1.5) = (-0.25) - (-0.75) = -0.25 + 0.75 = 0.5

Term 7: i=7, P₇ to P₈:

x7 y8 - x8 y7 = (-0.5)(0) - (0)(0.5) = 0 - 0 = 0

Now, sum all these terms:

0 + 0.5 + 1.5 + 1.5 + 2.0 + 1.0 + 0.5 + 0 =

Let me add them step by step:

Start with 0.

Add 0.5: 0.5

Add 1.5: 2.0

Add 1.5: 3.5

Add 2.0: 5.5

Add 1.0: 6.5

Add 0.5: 7.0

Add 0: 7.0

So the total sum is 7.0. Then the area is half the absolute value of that, so 7.0 / 2 = 3.5.

But 3.5 is equal to 7/2. However, let me check if I did the calculations correctly because I might have made an error in arithmetic.

Let me verify each term again:

Term 0: 0. Correct.

Term 1: 1*0.5 - 1.5*0 = 0.5. Correct.

Term 2: 1.5*1.5 - 1.5*0.5. 1.5*1.5 = 2.25, 1.5*0.5=0.75. 2.25 - 0.75 = 1.5. Correct.

Term 3: 1.5*2 - 1*1.5. That's 3 - 1.5 = 1.5. Correct.

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

Term 5: 0*1.5 - (-0.5)*2. Wait, original term was x5 y6 - x6 y5. x5 is 0.0, y6 is 1.5. x6 is -0.5, y5 is 2.0. So term is 0*1.5 - (-0.5)*2.0 = 0 - (-1.0) = 1.0. Correct.

Term 6: x6 y7 - x7 y6. x6 is -0.5, y7 is 0.5. x7 is -0.5, y6 is 1.5. So (-0.5)(0.5) - (-0.5)(1.5) = (-0.25) - (-0.75) = 0.5. Correct.

Term 7: 0. Correct.

Total sum: 0 + 0.5 + 1.5 + 1.5 + 2 + 1 + 0.5 + 0 = 0.5 + 1.5 = 2; 2 + 1.5 = 3.5; 3.5 + 2 = 5.5; 5.5 + 1 = 6.5; 6.5 + 0.5 = 7.0. Yes, that's correct.

Therefore, area is 7/2 = 3.5, which is 7/2. But let me think, is that possible? Let me check with another approach to verify.

Alternatively, maybe I can divide the octagon into simpler shapes. Let me look at the coordinates again.

The coordinates are:

(0, 0), (1, 0), (1.5, 0.5), (1.5, 1.5), (1.0, 2.0), (0.0, 2.0), (-0.5, 1.5), (-0.5, 0.5), back to (0, 0).

Looking at this shape, perhaps I can break it into rectangles and triangles or something else. Alternatively, maybe it's symmetric?

Looking at the coordinates, it seems symmetric with respect to the vertical line x = 0.5? Let me check.

Wait, the coordinates are symmetric if you reflect across x = 0.5? Let's see:

Original points:

(0,0), (1,0), (1.5,0.5), (1.5,1.5), (1.0,2.0), (0.0,2.0), (-0.5,1.5), (-0.5,0.5), (0,0)

If we reflect across x=0.5, then each x-coordinate becomes 1 - x. Let me check:

Take (0, 0): reflects to (1, 0). Which is present.

(1, 0) reflects to (0, 0). Also present.

(1.5, 0.5) reflects to (1 - 1.5, 0.5) = (-0.5, 0.5). Which is present.

Similarly, (1.5, 1.5) reflects to (-0.5, 1.5), which is present.

(1.0, 2.0) reflects to (0.0, 2.0), which is present.

(0.0, 2.0) reflects to (1.0, 2.0), which is present.

(-0.5, 1.5) reflects to (1.5, 1.5), present.

(-0.5, 0.5) reflects to (1.5, 0.5), present.

So yes, the figure is symmetric with respect to the line x = 0.5. Therefore, maybe we can compute the area of one half and double it?

Alternatively, maybe split into simpler components.

Alternatively, since I have coordinates, maybe I can use another method. But since the shoelace formula already gave 7/2, but perhaps I should verify if that answer is correct. Let me check with another method.

Alternatively, think of the octagon as a combination of a central rectangle and four isosceles right triangles?

Wait, maybe not. Let me think again about how the octagon is shaped.

Looking at the coordinates, from (0,0) to (1,0) to (1.5, 0.5) to (1.5, 1.5) to (1, 2) to (0,2) to (-0.5, 1.5) to (-0.5, 0.5) back to (0,0). So if I imagine this shape, it's like a rectangle with some parts cut off? Or maybe it's a combination of different shapes.

Alternatively, since the coordinates are known, perhaps I can divide the octagon into parts whose areas are easier to calculate. Let me try.

Looking at the coordinates, the figure is symmetric across x=0.5. Let me compute the area of the right half (from x=0 to x=1.5) and double it?

Wait, maybe not. Alternatively, think of it as a larger square with some regions subtracted. Let me check.

Alternatively, since the shoelace formula gives 3.5, which is 7/2. But maybe converting that into a fraction? Let me check if 7/2 is the answer. However, let me see if there's another approach.

Alternatively, think of the octagon as a regular octagon but with some sides scaled. Wait, but since the sides aren't all the same, maybe not. However, the problem is that I don't know the regular octagon's area, but perhaps my answer 7/2 is correct. Wait, but let me check with another method.

Alternatively, since the coordinates are known, I can use the shoelace formula again but perhaps write them in a different way. Wait, maybe I made a mistake in assigning the sides. Wait, let me check again the coordinates.

Wait, starting from (0,0):

Moving to (1,0): that's side length 1, correct.

Then to (1.5, 0.5): the displacement is (0.5, 0.5). The length of this side is sqrt( (0.5)^2 + (0.5)^2 ) = sqrt(0.25 + 0.25) = sqrt(0.5) = √2/2. Correct. So that side is length √2/2. Good.

Then to (1.5, 1.5): displacement is (0, 1). Length 1, correct. Then to (1.0, 2.0): displacement is (-0.5, 0.5). Length sqrt( (-0.5)^2 + (0.5)^2 ) = same as before √(0.5) = √2/2. Then from (1.0, 2.0) to (0.0, 2.0): displacement (-1, 0), length 1. Then to (-0.5, 1.5): displacement (-0.5, -0.5), length √(0.25 + 0.25) = √0.5. Then to (-0.5, 0.5): displacement (0, -1), length 1. Then to (0,0): displacement (0.5, -0.5), length √0.5. So all sides are as required. Therefore, coordinates are correct.

Therefore, shoelace formula calculation is correct, area is 7/2. But let me check again, maybe I miscalculated.

Wait, but 7/2 is 3.5, but I need to make sure. Let me cross-verify with another approach. Let me compute the area by dividing into trapezoids or something.

Looking at the coordinates, perhaps breaking the octagon into simpler parts. For example, from the coordinates, the figure is symmetric across x=0.5. Let me compute the area of the right half (from x=0 to x=1.5) and double it.

Alternatively, consider the entire figure as a polygon with coordinates listed. Alternatively, think of it as a combination of rectangles and triangles.

Looking at the coordinates:

Starting from (0,0) to (1,0) to (1.5, 0.5) to (1.5, 1.5) to (1, 2.0) to (0,2.0) to (-0.5,1.5) to (-0.5, 0.5) back to (0,0).

If I imagine this, the figure has a sort of "stretched" shape. Let me try to divide it into parts.

First, the rectangle from (0,0) to (1,0) to (1,2) to (0,2). That would be a rectangle of width 1 and height 2, area 2. But our octagon doesn't include the entire rectangle because there are indentations.

Alternatively, the octagon includes parts of that rectangle but also some extra areas. Wait, no. Let me think again.

Alternatively, perhaps the area can be calculated by subtracting the areas of the triangles that are cut off from a larger square. Wait, but I need to know the larger square. Let me think.

Alternatively, maybe think of the octagon as a combination of a central rectangle and four isosceles right triangles on the sides? Wait, not sure.

Alternatively, since the coordinates are known, maybe use the shoelace formula again but with decimal values. Wait, but I already did that.

Alternatively, maybe I can use vectors and compute the area via another formula. Wait, since we have vectors representing the sides, maybe use the formula for area of a polygon given side vectors? Not sure.

Alternatively, think of the octagon as a polygon with sides that can be represented as vectors, and the area can be calculated via the magnitude of cross products? Wait, for a polygon, the area can be found using the cross products of consecutive vectors. Wait, but I think that's similar to the shoelace formula.

Alternatively, since I have coordinates, maybe I can use the shoelace formula again with fractions instead of decimals. Let me redo the shoelace formula using fractions to check.

Original coordinates:

P₀: (0, 0)

P₁: (1, 0)

P₂: (3/2, 1/2)

P₃: (3/2, 3/2)

P₄: (1, 2)

P₅: (0, 2)

P₆: (-1/2, 3/2)

P₇: (-1/2, 1/2)

P₈: (0, 0)

So the shoelace formula terms:

Term 0: x0 y1 - x1 y0 = 0*0 - 1*0 = 0

Term 1: x1 y2 - x2 y1 = 1*(1/2) - (3/2)*0 = 1/2 - 0 = 1/2

Term 2: x2 y3 - x3 y2 = (3/2)*(3/2) - (3/2)*(1/2) = 9/4 - 3/4 = 6/4 = 3/2

Term 3: x3 y4 - x4 y3 = (3/2)*2 - 1*(3/2) = 3 - 3/2 = 3/2

Term 4: x4 y5 - x5 y4 = 1*2 - 0*2 = 2 - 0 = 2

Term 5: x5 y6 - x6 y5 = 0*(3/2) - (-1/2)*2 = 0 - (-1) = 1

Term 6: x6 y7 - x7 y6 = (-1/2)*(1/2) - (-1/2)*(3/2) = (-1/4) - (-3/4) = (-1/4 + 3/4) = 2/4 = 1/2

Term 7: x7 y8 - x8 y7 = (-1/2)*0 - 0*(1/2) = 0 - 0 = 0

Total sum: 0 + 1/2 + 3/2 + 3/2 + 2 + 1 + 1/2 + 0 = Let's compute:

Convert all to halves:

0 + 1/2 + 3/2 + 3/2 + 4/2 + 2/2 + 1/2 + 0 =

Adding numerators: 0 + 1 + 3 + 3 + 4 + 2 + 1 + 0 = 14. So total sum is 14/2 = 7. Then area is 7/2. Yes, same result. Therefore, 7/2 is correct.

But wait, the problem says the octagon has four sides of length 1 and four sides of length √2/2. But when I calculated the area via coordinates, I got 7/2. But is this answer reasonable? Let me check with a regular octagon.

Wait, what's the area of a regular octagon with side length a? The formula is 2(1 + √2)a². But in our case, the sides are not all equal, so it's different. However, if I consider the regular octagon that would have sides of some length, but our octagon may be similar? Not sure.

Alternatively, let me think about the coordinates. The maximum x-coordinate is 1.5, minimum is -0.5. Similarly, maximum y is 2.0, minimum is 0. So the bounding box is from x=-0.5 to 1.5 (width 2.0) and y=0 to 2.0 (height 2.0). So area of bounding box is 2.0 * 2.0 = 4.0. But our octagon's area is 3.5, which is less than 4, which makes sense.

Alternatively, if I calculate the area of the bounding box minus the areas of the four corners that are cut off. Let me see.

Looking at the coordinates, the octagon has four "notches" or cut-off corners. Each notch is a right triangle? Let me check.

Looking at the top right corner: the bounding box goes to (1.5, 2.0), but the octagon has a vertex at (1.0, 2.0) and (1.5, 1.5), so the corner is cut off by a triangle between (1.5, 2.0), (1.0, 2.0), (1.5, 1.5). Similarly for other corners.

Similarly on the top left: the bounding box goes to (-0.5, 2.0), but the octagon has (-0.5, 1.5) and (0, 2.0). So a triangle between (-0.5, 2.0), (0, 2.0), (-0.5, 1.5).

Same with bottom right and bottom left corners. So each of these four corners is a right triangle with legs of length 0.5 (since from x=1.0 to x=1.5 is 0.5, and from y=1.5 to y=2.0 is 0.5). Therefore, each triangle has area (0.5 * 0.5)/2 = 0.125. Four triangles would total 0.5. Therefore, area of octagon would be area of bounding box minus 0.5: 4 - 0.5 = 3.5. Which is 7/2.

Yes! So that confirms the shoelace result. Therefore, the area is indeed 7/2, which is 3.5. Therefore, the answer is 7/2.

But wait, let me check again. The bounding box area is 2.0 (width) * 2.0 (height) = 4.0. Then, each corner triangle is a right triangle with legs of 0.5. So area is (0.5 * 0.5)/2 = 0.125. Four triangles: 0.5. Therefore, 4 - 0.5 = 3.5. Yes, exactly. Therefore, this method also gives 7/2. Therefore, I'm confident that 7/2 is the correct area.

Therefore, the area of the octagon is 7/2, which is 3.5. So the answer is \boxed{\dfrac{7}{2}}.

Wait, but before finalizing, let me check again the problem statement to ensure I didn't misread anything. It says an equiangular octagon with four sides of length 1 and four sides of length √2/2, arranged so that no two consecutive sides have the same length. Which is exactly how I arranged them, alternating between 1 and √2/2. Therefore, everything seems correct.

Alternatively, another way to think about it: since the octagon is equiangular, it can be thought of as a rectangle with its corners cut off at 45 degrees, but with different side lengths. However, in this case, the side lengths are not all the same. However, in the case of a regular octagon, all sides are same, but in our case, sides alternate between two lengths. However, through the calculation, the area came out to 7/2, and verified through multiple methods.

Therefore, I think the answer is 7/2. So I will go with that.

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

To determine the area of the given equiangular octagon, we first extract and interpret the key information from the problem:

- **Equiangular**: This means all internal angles are equal. For a regular octagon, the internal angle is $135^\circ$, which applies here as well, even though the sides are not all equal.
- **Side lengths**: There are **four sides of length $1$** and **four sides of length $\frac{\sqrt{2}}{2}$**.
- **Arrangement**: No two consecutive sides have the same length. Thus, the sides must alternate between the two lengths.

---

### Step 1: Model the Octagon Using Coordinates

Since the octagon is equiangular and we know the side lengths and their alternating pattern, we can assign each side as a vector in the plane. Starting from the origin $(0,0)$, we can construct the vertices by summing up the side vectors in order, rotating by $45^\circ$ for each side (as the external angle is $45^\circ$), with alternating side lengths.

Specifically, the side vectors are as follows:

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

We compute the x and y components of each side vector and then sum them to get the coordinates of the vertices.

The resulting coordinates (in order) are:

- $(0, 0)$
- $(1, 0)$
- $(1.5, 0.5)$
- $(1.5, 1.5)$
- $(1, 2)$
- $(0, 2)$
- $(-0.5, 1.5)$
- $(-0.5, 0.5)$
- Back to $(0, 0)$

---

### Step 2: Apply the Shoelace Formula

The **Shoelace Formula** gives the area of a polygon with vertices $(x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)$. It is:

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

Applying this formula to the coordinates:

$$
\begin{aligned}
& (0)(0) + (1)(0.5) + (1.5)(1.5) + (1.5)(2) + (1)(2) + (0)(1.5) + (-0.5)(0.5) + (-0.5)(0) \\
& - [ (0)(1) + (0)(1.5) + (0.5)(1.5) + (1.5)(1) + (2)(-0.5) + (2)(-0.5) + (1.5)(-0.5) + (0.5)(0) ] \\
& = 0 + 0.5 + 2.25 + 3 + 2 + 0 - [ 0 + 0 + 0.75 + 1.5 - 1 - 1 - 0.75 + 0 ] \\
& = 0.5 + 2.25 + 3 + 2 = 7.75 \\
& = 7.75 - (0.75 + 1.5 + 1 + 1 - 0.75) = 7.75 - 3.5 = 4.25 \text{ (total)} \\
\end{aligned}
$$

Then, the area is:

$$
\frac{1}{2} \cdot (7.0) = \frac{7}{2}
$$

---

### Step 3: Final Answer

Thus, the area of the equiangular octagon 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. Also, no two consecutive sides have the same length. Let me try to visualize this first. An equiangular octagon means all the internal angles are equal. For a regular octagon, all sides and angles are equal, but this one is not regular since the sides are different. However, since it's equiangular, the angles are still all the same, which I think are 135 degrees each because in a regular octagon each internal angle is 135 degrees.

So, first, maybe I should recall the formula for the area of a regular octagon. But since this one is not regular, I can't directly use that formula. Maybe I can think of it as a regular octagon that's been stretched or something? But I need to figure out how the sides are arranged. The problem says there are 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 pattern of sides alternates between 1 and √2/2? Or maybe some other pattern?

Wait, there are four of each. Since no two consecutive sides can be the same, the sides must alternate between the two lengths. So maybe the pattern is 1, √2/2, 1, √2/2, etc. So the sides alternate between 1 and √2/2. Since it's an octagon, there are eight sides. So starting with 1, then √2/2, then 1, then √2/2, and so on. That would give four of each, and no two consecutive sides are the same. So that seems like a possible arrangement.

Now, if that's the case, maybe I can model this octagon as a modified regular octagon. Maybe by cutting off the corners of a square or something? Wait, but equiangular octagons can be thought of as similar to regular octagons but with sides of different lengths. Alternatively, maybe I can use coordinates. Let me think.

Since the octagon is equiangular, all the internal angles are 135 degrees. If I can model the octagon on a coordinate system, maybe using vectors or something. Let me try to think about how to compute the area.

Alternatively, maybe I can divide the octagon into simpler shapes like triangles, rectangles, or trapezoids. But since it's equiangular, perhaps there's some symmetry?

Wait, another approach: For an equiangular polygon, even if it's not regular, the sides can be related through vectors. Because each turn between sides is the same. For an octagon, each external angle is 45 degrees, since the sum of external angles is 360 degrees, so 360/8 = 45 degrees. Wait, but internal angles are 135, so external angles are 45. So when you walk around the octagon, at each vertex you turn 45 degrees.

If I think of the octagon as being constructed by vectors in the plane, each subsequent side is a vector obtained by rotating the previous one by 45 degrees. Wait, but the lengths of the sides are different. So maybe I can model the sides as vectors with angles 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, right? But since it's a closed polygon, the sum of all the vectors (sides) should be zero.

Therefore, if I denote the sides as vectors, each rotated by 45 degrees from the previous one, with lengths alternating between 1 and √2/2. Let me try to formalize this.

Let me denote the sides as vectors. Let me suppose that the first side is along the positive x-axis, length 1. Then the next side is at angle 45°, length √2/2. Then the next side is at angle 90°, length 1, then 135°, length √2/2, 180°, length 1, 225°, length √2/2, 270°, length 1, 315°, length √2/2, and then finally the last side should bring us back to the starting point. Wait, but there are 8 sides, so let me check:

Starting at angle 0°, then each subsequent side is rotated by 45°. So the angles of each side would be 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and then the eighth side would be at 315°, and then adding the eighth side should complete the polygon.

But let me check if that works. Let me denote the sides as vectors:

Let me define each side as a vector. Let me denote the sides as follows:

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°

Then, if we add up all these vectors, they should sum to zero. Let me check that. If this is the case, then the polygon closes.

So, let's compute the sum of these vectors.

First, let me write each vector in terms of components:

s₁ = 1*(cos 0°, sin 0°) = (1, 0)

s₂ = (√2/2)*(cos 45°, sin 45°) = (√2/2)*(√2/2, √2/2) = ( (√2 * √2)/4 , same ) = ( (2)/4, 2/4 ) = (0.5, 0.5)

Similarly, s₃ = 1*(cos 90°, sin 90°) = (0, 1)

s₄ = (√2/2)*(cos 135°, sin 135°) = (√2/2)*(-√2/2, √2/2) = (- (2)/4, (2)/4 ) = (-0.5, 0.5)

s₅ = 1*(cos 180°, sin 180°) = (-1, 0)

s₆ = (√2/2)*(cos 225°, sin 225°) = (√2/2)*(-√2/2, -√2/2) = (- (2)/4, - (2)/4 ) = (-0.5, -0.5)

s₇ = 1*(cos 270°, sin 270°) = (0, -1)

s₈ = (√2/2)*(cos 315°, sin 315°) = (√2/2)*(√2/2, -√2/2) = ( (2)/4, - (2)/4 ) = (0.5, -0.5)

Now let me sum up all these components:

First, sum of x-components:

s₁x + s₂x + s₃x + s₄x + s₅x + s₆x + s₇x + s₈x

= 1 + 0.5 + 0 + (-0.5) + (-1) + (-0.5) + 0 + 0.5

Compute step by step:

Start with 1.

Add 0.5: 1.5

Add 0: 1.5

Add (-0.5): 1.0

Add (-1): 0.0

Add (-0.5): -0.5

Add 0: -0.5

Add 0.5: 0.0

So total x-component sum is 0.

Similarly, y-components:

s₁y + s₂y + s₃y + s₄y + s₅y + s₆y + s₇y + s₈y

= 0 + 0.5 + 1 + 0.5 + 0 + (-0.5) + (-1) + (-0.5)

Compute step by step:

Start with 0.

Add 0.5: 0.5

Add 1: 1.5

Add 0.5: 2.0

Add 0: 2.0

Add (-0.5): 1.5

Add (-1): 0.5

Add (-0.5): 0.0

So y-component sum is also 0.

Therefore, the vectors do sum to zero. Therefore, this arrangement of sides actually forms a closed octagon. So that must be the correct configuration.

Therefore, the octagon can be represented by these vectors. Now, to compute its area. How can I compute the area of such a polygon?

One method to compute the area of a polygon given its vertices is the shoelace formula. For that, I need coordinates of the vertices. 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). Then, each subsequent vertex is obtained by adding the respective side vector.

Let me denote the vertices as P₀, P₁, P₂, ..., P₈, where P₀ is the starting point, and P₈ coincides with P₀ since it's a closed polygon.

Starting at P₀ = (0, 0).

Then:

P₁ = P₀ + s₁ = (0 + 1, 0 + 0) = (1, 0)

P₂ = P₁ + s₂ = (1 + 0.5, 0 + 0.5) = (1.5, 0.5)

P₃ = P₂ + s₃ = (1.5 + 0, 0.5 + 1) = (1.5, 1.5)

P₄ = P₃ + s₄ = (1.5 + (-0.5), 1.5 + 0.5) = (1.0, 2.0)

P₅ = P₄ + s₅ = (1.0 + (-1), 2.0 + 0) = (0.0, 2.0)

P₆ = P₅ + s₆ = (0.0 + (-0.5), 2.0 + (-0.5)) = (-0.5, 1.5)

P₇ = P₆ + s₇ = (-0.5 + 0, 1.5 + (-1)) = (-0.5, 0.5)

P₈ = P₇ + s₈ = (-0.5 + 0.5, 0.5 + (-0.5)) = (0.0, 0.0)

Which brings us back to P₀. Perfect, that works.

So now, the coordinates of the vertices are:

P₀: (0, 0)

P₁: (1, 0)

P₂: (1.5, 0.5)

P₃: (1.5, 1.5)

P₄: (1.0, 2.0)

P₅: (0.0, 2.0)

P₆: (-0.5, 1.5)

P₇: (-0.5, 0.5)

P₈: (0, 0) (which is same as P₀)

Now, I can apply the shoelace formula to compute the area. The shoelace formula says that the area is half the absolute value of the sum over the vertices of (x_i y_{i+1} - x_{i+1} y_i).

Let me write down the coordinates again:

Let me list them in order:

P₀: (0, 0)

P₁: (1, 0)

P₂: (1.5, 0.5)

P₃: (1.5, 1.5)

P₄: (1.0, 2.0)

P₅: (0.0, 2.0)

P₆: (-0.5, 1.5)

P₇: (-0.5, 0.5)

P₈: (0, 0)

So, applying the shoelace formula:

Compute sum_{i=0}^7 (x_i y_{i+1} - x_{i+1} y_i)

Let me compute each term step by step.

Term 0: i=0, P₀ to P₁:

x0 y1 - x1 y0 = (0)(0) - (1)(0) = 0 - 0 = 0

Term 1: i=1, P₁ to P₂:

x1 y2 - x2 y1 = (1)(0.5) - (1.5)(0) = 0.5 - 0 = 0.5

Term 2: i=2, P₂ to P₃:

x2 y3 - x3 y2 = (1.5)(1.5) - (1.5)(0.5) = 2.25 - 0.75 = 1.5

Term 3: i=3, P₃ to P₄:

x3 y4 - x4 y3 = (1.5)(2.0) - (1.0)(1.5) = 3.0 - 1.5 = 1.5

Term 4: i=4, P₄ to P₅:

x4 y5 - x5 y4 = (1.0)(2.0) - (0.0)(2.0) = 2.0 - 0 = 2.0

Term 5: i=5, P₅ to P₆:

x5 y6 - x6 y5 = (0.0)(1.5) - (-0.5)(2.0) = 0 - (-1.0) = 1.0

Term 6: i=6, P₆ to P₇:

x6 y7 - x7 y6 = (-0.5)(0.5) - (-0.5)(1.5) = (-0.25) - (-0.75) = -0.25 + 0.75 = 0.5

Term 7: i=7, P₇ to P₈:

x7 y8 - x8 y7 = (-0.5)(0) - (0)(0.5) = 0 - 0 = 0

Now, sum all these terms:

0 + 0.5 + 1.5 + 1.5 + 2.0 + 1.0 + 0.5 + 0 =

Let me add them step by step:

Start with 0.

Add 0.5: 0.5

Add 1.5: 2.0

Add 1.5: 3.5

Add 2.0: 5.5

Add 1.0: 6.5

Add 0.5: 7.0

Add 0: 7.0

So the total sum is 7.0. Then the area is half the absolute value of that, so 7.0 / 2 = 3.5.

But 3.5 is equal to 7/2. However, let me check if I did the calculations correctly because I might have made an error in arithmetic.

Let me verify each term again:

Term 0: 0. Correct.

Term 1: 1*0.5 - 1.5*0 = 0.5. Correct.

Term 2: 1.5*1.5 - 1.5*0.5. 1.5*1.5 = 2.25, 1.5*0.5=0.75. 2.25 - 0.75 = 1.5. Correct.

Term 3: 1.5*2 - 1*1.5. That's 3 - 1.5 = 1.5. Correct.

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

Term 5: 0*1.5 - (-0.5)*2. Wait, original term was x5 y6 - x6 y5. x5 is 0.0, y6 is 1.5. x6 is -0.5, y5 is 2.0. So term is 0*1.5 - (-0.5)*2.0 = 0 - (-1.0) = 1.0. Correct.

Term 6: x6 y7 - x7 y6. x6 is -0.5, y7 is 0.5. x7 is -0.5, y6 is 1.5. So (-0.5)(0.5) - (-0.5)(1.5) = (-0.25) - (-0.75) = 0.5. Correct.

Term 7: 0. Correct.

Total sum: 0 + 0.5 + 1.5 + 1.5 + 2 + 1 + 0.5 + 0 = 0.5 + 1.5 = 2; 2 + 1.5 = 3.5; 3.5 + 2 = 5.5; 5.5 + 1 = 6.5; 6.5 + 0.5 = 7.0. Yes, that's correct.

Therefore, area is 7/2 = 3.5, which is 7/2. But let me think, is that possible? Let me check with another approach to verify.

Alternatively, maybe I can divide the octagon into simpler shapes. Let me look at the coordinates again.

The coordinates are:

(0, 0), (1, 0), (1.5, 0.5), (1.5, 1.5), (1.0, 2.0), (0.0, 2.0), (-0.5, 1.5), (-0.5, 0.5), back to (0, 0).

Looking at this shape, perhaps I can break it into rectangles and triangles or something else. Alternatively, maybe it's symmetric?

Looking at the coordinates, it seems symmetric with respect to the vertical line x = 0.5? Let me check.

Wait, the coordinates are symmetric if you reflect across x = 0.5? Let's see:

Original points:

(0,0), (1,0), (1.5,0.5), (1.5,1.5), (1.0,2.0), (0.0,2.0), (-0.5,1.5), (-0.5,0.5), (0,0)

If we reflect across x=0.5, then each x-coordinate becomes 1 - x. Let me check:

Take (0, 0): reflects to (1, 0). Which is present.

(1, 0) reflects to (0, 0). Also present.

(1.5, 0.5) reflects to (1 - 1.5, 0.5) = (-0.5, 0.5). Which is present.

Similarly, (1.5, 1.5) reflects to (-0.5, 1.5), which is present.

(1.0, 2.0) reflects to (0.0, 2.0), which is present.

(0.0, 2.0) reflects to (1.0, 2.0), which is present.

(-0.5, 1.5) reflects to (1.5, 1.5), present.

(-0.5, 0.5) reflects to (1.5, 0.5), present.

So yes, the figure is symmetric with respect to the line x = 0.5. Therefore, maybe we can compute the area of one half and double it?

Alternatively, maybe split into simpler components.

Alternatively, since I have coordinates, maybe I can use another method. But since the shoelace formula already gave 7/2, but perhaps I should verify if that answer is correct. Let me check with another method.

Alternatively, think of the octagon as a combination of a central rectangle and four isosceles right triangles?

Wait, maybe not. Let me think again about how the octagon is shaped.

Looking at the coordinates, from (0,0) to (1,0) to (1.5, 0.5) to (1.5, 1.5) to (1, 2) to (0,2) to (-0.5, 1.5) to (-0.5, 0.5) back to (0,0). So if I imagine this shape, it's like a rectangle with some parts cut off? Or maybe it's a combination of different shapes.

Alternatively, since the coordinates are known, perhaps I can divide the octagon into parts whose areas are easier to calculate. Let me try.

Looking at the coordinates, the figure is symmetric across x=0.5. Let me compute the area of the right half (from x=0 to x=1.5) and double it?

Wait, maybe not. Alternatively, think of it as a larger square with some regions subtracted. Let me check.

Alternatively, since the shoelace formula gives 3.5, which is 7/2. But maybe converting that into a fraction? Let me check if 7/2 is the answer. However, let me see if there's another approach.

Alternatively, think of the octagon as a regular octagon but with some sides scaled. Wait, but since the sides aren't all the same, maybe not. However, the problem is that I don't know the regular octagon's area, but perhaps my answer 7/2 is correct. Wait, but let me check with another method.

Alternatively, since the coordinates are known, I can use the shoelace formula again but perhaps write them in a different way. Wait, maybe I made a mistake in assigning the sides. Wait, let me check again the coordinates.

Wait, starting from (0,0):

Moving to (1,0): that's side length 1, correct.

Then to (1.5, 0.5): the displacement is (0.5, 0.5). The length of this side is sqrt( (0.5)^2 + (0.5)^2 ) = sqrt(0.25 + 0.25) = sqrt(0.5) = √2/2. Correct. So that side is length √2/2. Good.

Then to (1.5, 1.5): displacement is (0, 1). Length 1, correct. Then to (1.0, 2.0): displacement is (-0.5, 0.5). Length sqrt( (-0.5)^2 + (0.5)^2 ) = same as before √(0.5) = √2/2. Then from (1.0, 2.0) to (0.0, 2.0): displacement (-1, 0), length 1. Then to (-0.5, 1.5): displacement (-0.5, -0.5), length √(0.25 + 0.25) = √0.5. Then to (-0.5, 0.5): displacement (0, -1), length 1. Then to (0,0): displacement (0.5, -0.5), length √0.5. So all sides are as required. Therefore, coordinates are correct.

Therefore, shoelace formula calculation is correct, area is 7/2. But let me check again, maybe I miscalculated.

Wait, but 7/2 is 3.5, but I need to make sure. Let me cross-verify with another approach. Let me compute the area by dividing into trapezoids or something.

Looking at the coordinates, perhaps breaking the octagon into simpler parts. For example, from the coordinates, the figure is symmetric across x=0.5. Let me compute the area of the right half (from x=0 to x=1.5) and double it.

Alternatively, consider the entire figure as a polygon with coordinates listed. Alternatively, think of it as a combination of rectangles and triangles.

Looking at the coordinates:

Starting from (0,0) to (1,0) to (1.5, 0.5) to (1.5, 1.5) to (1, 2.0) to (0,2.0) to (-0.5,1.5) to (-0.5, 0.5) back to (0,0).

If I imagine this, the figure has a sort of "stretched" shape. Let me try to divide it into parts.

First, the rectangle from (0,0) to (1,0) to (1,2) to (0,2). That would be a rectangle of width 1 and height 2, area 2. But our octagon doesn't include the entire rectangle because there are indentations.

Alternatively, the octagon includes parts of that rectangle but also some extra areas. Wait, no. Let me think again.

Alternatively, perhaps the area can be calculated by subtracting the areas of the triangles that are cut off from a larger square. Wait, but I need to know the larger square. Let me think.

Alternatively, maybe think of the octagon as a combination of a central rectangle and four isosceles right triangles on the sides? Wait, not sure.

Alternatively, since the coordinates are known, maybe use the shoelace formula again but with decimal values. Wait, but I already did that.

Alternatively, maybe I can use vectors and compute the area via another formula. Wait, since we have vectors representing the sides, maybe use the formula for area of a polygon given side vectors? Not sure.

Alternatively, think of the octagon as a polygon with sides that can be represented as vectors, and the area can be calculated via the magnitude of cross products? Wait, for a polygon, the area can be found using the cross products of consecutive vectors. Wait, but I think that's similar to the shoelace formula.

Alternatively, since I have coordinates, maybe I can use the shoelace formula again with fractions instead of decimals. Let me redo the shoelace formula using fractions to check.

Original coordinates:

P₀: (0, 0)

P₁: (1, 0)

P₂: (3/2, 1/2)

P₃: (3/2, 3/2)

P₄: (1, 2)

P₅: (0, 2)

P₆: (-1/2, 3/2)

P₇: (-1/2, 1/2)

P₈: (0, 0)

So the shoelace formula terms:

Term 0: x0 y1 - x1 y0 = 0*0 - 1*0 = 0

Term 1: x1 y2 - x2 y1 = 1*(1/2) - (3/2)*0 = 1/2 - 0 = 1/2

Term 2: x2 y3 - x3 y2 = (3/2)*(3/2) - (3/2)*(1/2) = 9/4 - 3/4 = 6/4 = 3/2

Term 3: x3 y4 - x4 y3 = (3/2)*2 - 1*(3/2) = 3 - 3/2 = 3/2

Term 4: x4 y5 - x5 y4 = 1*2 - 0*2 = 2 - 0 = 2

Term 5: x5 y6 - x6 y5 = 0*(3/2) - (-1/2)*2 = 0 - (-1) = 1

Term 6: x6 y7 - x7 y6 = (-1/2)*(1/2) - (-1/2)*(3/2) = (-1/4) - (-3/4) = (-1/4 + 3/4) = 2/4 = 1/2

Term 7: x7 y8 - x8 y7 = (-1/2)*0 - 0*(1/2) = 0 - 0 = 0

Total sum: 0 + 1/2 + 3/2 + 3/2 + 2 + 1 + 1/2 + 0 = Let's compute:

Convert all to halves:

0 + 1/2 + 3/2 + 3/2 + 4/2 + 2/2 + 1/2 + 0 =

Adding numerators: 0 + 1 + 3 + 3 + 4 + 2 + 1 + 0 = 14. So total sum is 14/2 = 7. Then area is 7/2. Yes, same result. Therefore, 7/2 is correct.

But wait, the problem says the octagon has four sides of length 1 and four sides of length √2/2. But when I calculated the area via coordinates, I got 7/2. But is this answer reasonable? Let me check with a regular octagon.

Wait, what's the area of a regular octagon with side length a? The formula is 2(1 + √2)a². But in our case, the sides are not all equal, so it's different. However, if I consider the regular octagon that would have sides of some length, but our octagon may be similar? Not sure.

Alternatively, let me think about the coordinates. The maximum x-coordinate is 1.5, minimum is -0.5. Similarly, maximum y is 2.0, minimum is 0. So the bounding box is from x=-0.5 to 1.5 (width 2.0) and y=0 to 2.0 (height 2.0). So area of bounding box is 2.0 * 2.0 = 4.0. But our octagon's area is 3.5, which is less than 4, which makes sense.

Alternatively, if I calculate the area of the bounding box minus the areas of the four corners that are cut off. Let me see.

Looking at the coordinates, the octagon has four "notches" or cut-off corners. Each notch is a right triangle? Let me check.

Looking at the top right corner: the bounding box goes to (1.5, 2.0), but the octagon has a vertex at (1.0, 2.0) and (1.5, 1.5), so the corner is cut off by a triangle between (1.5, 2.0), (1.0, 2.0), (1.5, 1.5). Similarly for other corners.

Similarly on the top left: the bounding box goes to (-0.5, 2.0), but the octagon has (-0.5, 1.5) and (0, 2.0). So a triangle between (-0.5, 2.0), (0, 2.0), (-0.5, 1.5).

Same with bottom right and bottom left corners. So each of these four corners is a right triangle with legs of length 0.5 (since from x=1.0 to x=1.5 is 0.5, and from y=1.5 to y=2.0 is 0.5). Therefore, each triangle has area (0.5 * 0.5)/2 = 0.125. Four triangles would total 0.5. Therefore, area of octagon would be area of bounding box minus 0.5: 4 - 0.5 = 3.5. Which is 7/2.

Yes! So that confirms the shoelace result. Therefore, the area is indeed 7/2, which is 3.5. Therefore, the answer is 7/2.

But wait, let me check again. The bounding box area is 2.0 (width) * 2.0 (height) = 4.0. Then, each corner triangle is a right triangle with legs of 0.5. So area is (0.5 * 0.5)/2 = 0.125. Four triangles: 0.5. Therefore, 4 - 0.5 = 3.5. Yes, exactly. Therefore, this method also gives 7/2. Therefore, I'm confident that 7/2 is the correct area.

Therefore, the area of the octagon is 7/2, which is 3.5. So the answer is \boxed{\dfrac{7}{2}}.

Wait, but before finalizing, let me check again the problem statement to ensure I didn't misread anything. It says an equiangular octagon with four sides of length 1 and four sides of length √2/2, arranged so that no two consecutive sides have the same length. Which is exactly how I arranged them, alternating between 1 and √2/2. Therefore, everything seems correct.

Alternatively, another way to think about it: since the octagon is equiangular, it can be thought of as a rectangle with its corners cut off at 45 degrees, but with different side lengths. However, in this case, the side lengths are not all the same. However, in the case of a regular octagon, all sides are same, but in our case, sides alternate between two lengths. However, through the calculation, the area came out to 7/2, and verified through multiple methods.

Therefore, I think the answer is 7/2. So I will go with that.

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

To determine the area of the given equiangular octagon, we first extract and interpret the key information from the problem:

- **Equiangular**: This means all internal angles are equal. For a regular octagon, the internal angle is $135^\circ$, which applies here as well, even though the sides are not all equal.
- **Side lengths**: There are **four sides of length $1$** and **four sides of length $\frac{\sqrt{2}}{2}$**.
- **Arrangement**: No two consecutive sides have the same length. Thus, the sides must alternate between the two lengths.

---

### Step 1: Model the Octagon Using Coordinates

Since the octagon is equiangular and we know the side lengths and their alternating pattern, we can assign each side as a vector in the plane. Starting from the origin $(0,0)$, we can construct the vertices by summing up the side vectors in order, rotating by $45^\circ$ for each side (as the external angle is $45^\circ$), with alternating side lengths.

Specifically, the side vectors are as follows:

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

We compute the x and y components of each side vector and then sum them to get the coordinates of the vertices.

The resulting coordinates (in order) are:

- $(0, 0)$
- $(1, 0)$
- $(1.5, 0.5)$
- $(1.5, 1.5)$
- $(1, 2)$
- $(0, 2)$
- $(-0.5, 1.5)$
- $(-0.5, 0.5)$
- Back to $(0, 0)$

---

### Step 2: Apply the Shoelace Formula

The **Shoelace Formula** gives the area of a polygon with vertices $(x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)$. It is:

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

Applying this formula to the coordinates:

$$
\begin{aligned}
& (0)(0) + (1)(0.5) + (1.5)(1.5) + (1.5)(2) + (1)(2) + (0)(1.5) + (-0.5)(0.5) + (-0.5)(0) \\
& - [ (0)(1) + (0)(1.5) + (0.5)(1.5) + (1.5)(1) + (2)(-0.5) + (2)(-0.5) + (1.5)(-0.5) + (0.5)(0) ] \\
& = 0 + 0.5 + 2.25 + 3 + 2 + 0 - [ 0 + 0 + 0.75 + 1.5 - 1 - 1 - 0.75 + 0 ] \\
& = 0.5 + 2.25 + 3 + 2 = 7.75 \\
& = 7.75 - (0.75 + 1.5 + 1 + 1 - 0.75) = 7.75 - 3.5 = 4.25 \text{ (total)} \\
\end{aligned}
$$

Then, the area is:

$$
\frac{1}{2} \cdot (7.0) = \frac{7}{2}
$$

---

### Step 3: Final Answer

Thus, the area of the equiangular octagon 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, let me try to figure out this problem. So, there's a sequence defined by the recurrence relation $ a_{i + 1} = \frac{1}{1 - a_i} $ for $ i \geq 1 $. We're told that $ a_3 = a_1 $, and we need to compute $ (a_9)^9 $. Hmm, let me break this down step by step.

First, let me recall what a recurrence relation is. It's a way to define a sequence where each term is defined as a function of the preceding terms. In this case, each term is defined based on the previous term. Specifically, each term is the reciprocal of $ 1 - $ the previous term. So, to get $ a_2 $, I use $ a_1 $, to get $ a_3 $, I use $ a_2 $, and so on.

But we have a condition here: $ a_3 = a_1 $. That means that when we apply the recurrence relation twice, starting from $ a_1 $, we end up back at $ a_1 $. So, maybe this sequence is periodic with period 2? Or maybe some other period? Let me check.

Let me write down the terms in terms of $ a_1 $. Let's denote $ a_1 $ as $ x $ to make notation easier. Then,

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

Wait, let me compute $ a_3 $ step by step. Let me compute $ a_2 $ first. If $ a_2 = 1/(1 - x) $, then $ a_3 $ is 1 divided by (1 - a_2). So:

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

Let me simplify the denominator:

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

So, $ a_3 = \frac{1}{ (-x)/(1 - x) } = \frac{1 - x}{-x} = \frac{x - 1}{x} $

Therefore, $ a_3 = \frac{x - 1}{x} $. But we are told that $ a_3 = a_1 $, which is $ x $. Therefore, we can set up the equation:

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

Let me solve this equation for $ x $. Multiply both sides by $ x $:

$ x - 1 = x^2 $

Rearranging:

$ x^2 - x + 1 = 0 $

Wait, hold on. Let me check my algebra again. If I have $ \frac{x - 1}{x} = x $, then multiplying both sides by $ x $:

Left side: $ x - 1 $

Right side: $ x \times x = x^2 $

So the equation is:

$ x - 1 = x^2 $

Which can be rewritten as:

$ x^2 - x + 1 = 0 $

Wait, no. Wait, moving all terms to one side:

$ x^2 - x + 1 = 0 $? Wait, if we have $ x - 1 = x^2 $, then subtract $ x - 1 $ from both sides:

$ 0 = x^2 - x + 1 $

Yes, so $ x^2 - x + 1 = 0 $

Wait, but this quadratic equation... Let me check the discriminant. The discriminant $ D = (-1)^2 - 4 \times 1 \times 1 = 1 - 4 = -3 $. Oh! So the discriminant is negative. That would mean that there are no real solutions. But the problem didn't specify whether the sequence is real or complex. Wait, but maybe there's a mistake here?

Wait, maybe I made a mistake in calculating $ a_3 $. Let me check that again.

Starting with $ a_1 = x $, then:

$ a_2 = 1/(1 - x) $

Then $ a_3 = 1/(1 - a_2) = 1 / [1 - 1/(1 - x)] $

Let me compute the denominator:

$ 1 - 1/(1 - x) $. Let me write 1 as $ (1 - x)/(1 - x) $, so:

$ (1 - x)/(1 - x) - 1/(1 - x) = [ (1 - x) - 1 ] / (1 - x) = (-x)/(1 - x) $. Yes, so denominator is $ -x/(1 - x) $, so reciprocal is $ - (1 - x)/x = (x - 1)/x $. So that seems correct. Therefore, $ a_3 = (x - 1)/x $. Then setting equal to $ a_1 = x $, so:

$ (x - 1)/x = x $

Multiply both sides by x:

$ x - 1 = x^2 $

Which is $ x^2 - x + 1 = 0 $, discriminant negative. So complex roots?

But maybe the problem is okay with complex numbers? But the question says "compute $ (a_9)^9 $", but doesn't specify. However, maybe I made an error in assuming $ a_3 = a_1 $, so perhaps there's another way. Wait, maybe I need to check if there is a periodicity?

Wait, if $ a_3 = a_1 $, then maybe the sequence is periodic with period 2? So if $ a_1 = a_3 $, then $ a_2 = a_4 $, etc. Then, if the sequence is periodic with period 2, then $ a_9 $ would be equal to $ a_1 $, because 9 is odd. Wait, but 9 divided by 2 is 4 with remainder 1. So if the period is 2, then $ a_9 = a_1 $, so $ a_9^9 = (a_1)^9 $. But then we need to compute $ a_1 $, but since the quadratic equation has complex roots, maybe there is some specific value?

Wait, but maybe there is a mistake here. Maybe the problem is intended for real numbers? If there are no real solutions, then maybe the problem is ill-posed? But that seems unlikely. Maybe I made an error in the calculation?

Alternatively, maybe I need to consider that the sequence could have a longer period? Let's see. Suppose that $ a_3 = a_1 $, but maybe there's another relation?

Alternatively, perhaps there's a miscalculation here. Let me check again.

Let me try plugging in some numbers. Let me suppose that $ a_1 $ is such that $ a_3 = a_1 $.

Alternatively, perhaps I need to consider that applying the recurrence three times brings us back to the original value? Wait, but the problem says $ a_3 = a_1 $. So if we start with $ a_1 $, then $ a_2 = 1/(1 - a_1) $, $ a_3 = 1/(1 - a_2) $, and then $ a_3 = a_1 $. So the function mapping $ a_1 \to a_3 $ is the same as the function composed twice. So maybe the function f(x) = 1/(1 - x), then f(f(x)) = ?

Let me compute f(f(x)):

f(x) = 1/(1 - x)

f(f(x)) = 1 / [1 - f(x)] = 1 / [1 - 1/(1 - x)] as before, which simplifies to (x - 1)/x, which is what we had earlier. So f(f(x)) = (x - 1)/x.

So setting f(f(x)) = x gives (x - 1)/x = x, leading to x^2 - x + 1 = 0. Which is complex. So maybe complex numbers are allowed here?

But then, maybe even though the roots are complex, we can still compute $ a_9^9 $. Wait, but perhaps there's a different approach. Let me consider the sequence.

Since $ a_3 = a_1 $, then the sequence is periodic with period 2? Wait, not necessarily. Because if $ a_1 = a_3 $, then $ a_2 = a_4 $, $ a_3 = a_5 $, etc. So the period is 2. So then, since 9 is odd, $ a_9 = a_1 $. Therefore, $ a_9^9 = (a_1)^9 $. Therefore, if we can find $ a_1 $, then we can compute this value.

But since $ a_1 $ satisfies $ x^2 - x + 1 = 0 $, then $ x^2 = x - 1 $. Maybe we can use this to compute $ x^9 $?

Alternatively, perhaps we can find a pattern in the powers of x. Let me try that.

Given that $ x^2 = x - 1 $, then we can express higher powers of x in terms of lower powers. Let me compute:

Let me note that:

x^2 = x - 1

x^3 = x * x^2 = x(x - 1) = x^2 - x = (x - 1) - x = -1

Wait, really? Let me compute step by step:

x^2 = x - 1.

x^3 = x * x^2 = x*(x - 1) = x^2 - x. But x^2 is x - 1, so substitute:

x^3 = (x - 1) - x = -1. So x^3 = -1.

Interesting! Then x^3 = -1.

Then x^4 = x * x^3 = x*(-1) = -x

x^5 = x * x^4 = x*(-x) = -x^2 = -(x - 1) = -x + 1

x^6 = x * x^5 = x*(-x + 1) = -x^2 + x = -(x - 1) + x = -x + 1 + x = 1

Wait, x^6 = 1?

Then x^6 = 1. So x^6 = 1. Then x^9 = x^(6 + 3) = x^6 * x^3 = 1 * (-1) = -1. Therefore, x^9 = -1.

Therefore, if a1 = x, then a9 = x, so (a9)^9 = x^9 = -1. Therefore, the answer is -1?

But let me check this again. Let me retrace:

Starting from x^2 = x - 1.

x^3 = x * x^2 = x(x - 1) = x^2 - x = (x - 1) - x = -1. Correct.

x^3 = -1.

x^4 = x * x^3 = x*(-1) = -x.

x^5 = x * x^4 = x*(-x) = -x^2 = -(x - 1) = -x + 1.

x^6 = x * x^5 = x*(-x + 1) = -x^2 + x = -(x - 1) + x = -x + 1 + x = 1. Correct. So x^6 = 1.

Therefore x^6 = 1, so x^9 = x^6 * x^3 = 1 * (-1) = -1. Therefore, yes, (a_9)^9 = -1. So the answer is \boxed{-1}? Wait, but let me check again if this is consistent.

But let me also check if there are other possibilities. Since the roots of x^2 - x + 1 = 0 are complex numbers. Let me find them. The roots are [1 ± sqrt(-3)] / 2 = [1 ± i√3]/2. These are complex numbers on the unit circle. In fact, they are the primitive sixth roots of unity? Because x^6 = 1, so they are sixth roots of unity. Let me check. Let me compute x^6 for x = [1 + i√3]/2.

Wait, [1 + i√3]/2 is e^(iπ/3), right? Because cos(π/3) = 1/2, sin(π/3) = √3/2. Therefore, raising this to the 6th power: [e^(iπ/3)]^6 = e^(i2π) = 1. Similarly, [1 - i√3]/2 is e^(-iπ/3), and raising it to the 6th power gives 1 as well. So yes, x^6 = 1. Therefore, x is a primitive 6th root of unity? Well, since x^3 = -1, which is e^(iπ), so x^3 = -1, so x^6 = 1. Therefore, x is a primitive 6th root of unity. Therefore, x^6 = 1, so x^9 = x^(6 + 3) = x^6 * x^3 = 1 * (-1) = -1. So that's consistent. Therefore, regardless of which root we take, x^9 = -1. Therefore, (a_9)^9 = -1.

Therefore, the answer is \boxed{-1}?

But let me check once again if my assumption that the sequence is periodic with period 2 is correct. Given that a3 = a1, so a1 = a3, then a4 = 1/(1 - a3) = 1/(1 - a1) = a2, so a4 = a2. Then a5 = 1/(1 - a4) = 1/(1 - a2) = a3 = a1. Similarly, a6 = a4 = a2, a7 = a5 = a1, and so on. So indeed, the sequence is periodic with period 2. Therefore, odd terms are equal to a1, even terms equal to a2. Therefore, since 9 is odd, a9 = a1. Therefore, (a9)^9 = (a1)^9 = -1. So the answer is -1.

But let me check with actual numbers. Suppose I pick a complex number x such that x^2 - x + 1 = 0. Let me take x = [1 + i√3]/2. Then compute x^3, x^6, x^9.

First, x = e^(iπ/3). Then x^3 = e^(iπ) = -1. Then x^6 = e^(i2π) = 1. Then x^9 = e^(i3π) = e^(iπ) = -1. So indeed, x^9 = -1. Similarly for the other root, x = e^(-iπ/3), same result. Therefore, regardless of the root, x^9 is -1. Therefore, (a_9)^9 = -1.

Therefore, I think that the answer is \boxed{-1}. But wait, let me check if there's any mistake in assuming that a9 = a1. Since the sequence has period 2, then:

a1, a2, a1, a2, a1, a2, a1, a2, a1,...

So a9 is the 9th term. Since the first term is a1, then term 1: a1, term 2: a2, term 3: a1, term 4: a2, term 5: a1, term 6: a2, term 7: a1, term 8: a2, term 9: a1. Yes, so a9 = a1. Therefore, (a9)^9 = (a1)^9 = -1. Therefore, the answer is indeed -1.

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

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

We are given a recurrence relation for a sequence $(a_n)$ defined by:

$$
a_{i+1} = \frac{1}{1 - a_i}
$$

with the condition $a_3 = a_1$. The goal is to compute $(a_9)^9$.

---

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

Let $a_1 = x$. Then:

- $a_2 = \frac{1}{1 - x}$
- $a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - \frac{1}{1 - x}} = \frac{1}{\left( \frac{(1 - x) - 1}{1 - x} \right)} = \frac{1 - x}{-x} = \frac{x - 1}{x}$

We are told that $a_3 = a_1$. Therefore:

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

Multiply both sides by $x$ (assuming $x \ne 0$):

$$
x - 1 = x^2 \Rightarrow x^2 - x + 1 = 0
$$

---

### Step 2: Analyze the quadratic equation

The equation $x^2 - x + 1 = 0$ has a discriminant:

$$
D = (-1)^2 - 4(1)(1) = 1 - 4 = -3
$$

So the solutions are:

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

These are complex conjugate roots, and they are the **primitive 6th roots of unity** (since they satisfy $x^6 = 1$).

---

### Step 3: Determine the periodicity of the sequence

We observe the recurrence:

$$
a_{i+1} = \frac{1}{1 - a_i}
$$

Let’s define the function $f(x) = \frac{1}{1 - x}$. Then the sequence evolves as:

$$
a_1 = x \\
a_2 = f(x) \\
a_3 = f(f(x)) \\
a_4 = f(f(f(x))) = f(a_3) \\
\text{and so on}
$$

Given $a_3 = a_1$, we can conclude that the sequence is periodic with **period 2**:

$$
a_1 = a_3 = a_5 = a_7 = a_9 = \dots \\
a_2 = a_4 = a_6 = a_8 = \dots
$$

So, since 9 is odd, $a_9 = a_1 = x$.

---

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

Since $a_9 = a_1 = x$, we compute:

$$
(a_9)^9 = x^9
$$

We already established that the roots of $x^2 - x + 1 = 0$ are the primitive 6th roots of unity. Therefore, $x^6 = 1$. Thus:

$$
x^9 = x^{6 + 3} = x^6 \cdot x^3 = 1 \cdot x^3
$$

From earlier, we saw that:

$$
x^3 = \frac{1}{1 - \frac{1}{1 - x}} = \frac{x - 1}{x}
$$

But from the equation $x^2 = x - 1$, we can compute:

$$
x^3 = x \cdot x^2 = x(x - 1) = x^2 - x = (x - 1) - x = -1
$$

Therefore:

$$
x^9 = -1
$$

---

### ✅ Final Answer

$$
\boxed{-1}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, there's a sequence defined by the recurrence relation $ a_{i + 1} = \frac{1}{1 - a_i} $ for $ i \geq 1 $. We're told that $ a_3 = a_1 $, and we need to compute $ (a_9)^9 $. Hmm, let me break this down step by step.

First, let me recall what a recurrence relation is. It's a way to define a sequence where each term is defined as a function of the preceding terms. In this case, each term is defined based on the previous term. Specifically, each term is the reciprocal of $ 1 - $ the previous term. So, to get $ a_2 $, I use $ a_1 $, to get $ a_3 $, I use $ a_2 $, and so on.

But we have a condition here: $ a_3 = a_1 $. That means that when we apply the recurrence relation twice, starting from $ a_1 $, we end up back at $ a_1 $. So, maybe this sequence is periodic with period 2? Or maybe some other period? Let me check.

Let me write down the terms in terms of $ a_1 $. Let's denote $ a_1 $ as $ x $ to make notation easier. Then,

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

Wait, let me compute $ a_3 $ step by step. Let me compute $ a_2 $ first. If $ a_2 = 1/(1 - x) $, then $ a_3 $ is 1 divided by (1 - a_2). So:

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

Let me simplify the denominator:

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

So, $ a_3 = \frac{1}{ (-x)/(1 - x) } = \frac{1 - x}{-x} = \frac{x - 1}{x} $

Therefore, $ a_3 = \frac{x - 1}{x} $. But we are told that $ a_3 = a_1 $, which is $ x $. Therefore, we can set up the equation:

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

Let me solve this equation for $ x $. Multiply both sides by $ x $:

$ x - 1 = x^2 $

Rearranging:

$ x^2 - x + 1 = 0 $

Wait, hold on. Let me check my algebra again. If I have $ \frac{x - 1}{x} = x $, then multiplying both sides by $ x $:

Left side: $ x - 1 $

Right side: $ x \times x = x^2 $

So the equation is:

$ x - 1 = x^2 $

Which can be rewritten as:

$ x^2 - x + 1 = 0 $

Wait, no. Wait, moving all terms to one side:

$ x^2 - x + 1 = 0 $? Wait, if we have $ x - 1 = x^2 $, then subtract $ x - 1 $ from both sides:

$ 0 = x^2 - x + 1 $

Yes, so $ x^2 - x + 1 = 0 $

Wait, but this quadratic equation... Let me check the discriminant. The discriminant $ D = (-1)^2 - 4 \times 1 \times 1 = 1 - 4 = -3 $. Oh! So the discriminant is negative. That would mean that there are no real solutions. But the problem didn't specify whether the sequence is real or complex. Wait, but maybe there's a mistake here?

Wait, maybe I made a mistake in calculating $ a_3 $. Let me check that again.

Starting with $ a_1 = x $, then:

$ a_2 = 1/(1 - x) $

Then $ a_3 = 1/(1 - a_2) = 1 / [1 - 1/(1 - x)] $

Let me compute the denominator:

$ 1 - 1/(1 - x) $. Let me write 1 as $ (1 - x)/(1 - x) $, so:

$ (1 - x)/(1 - x) - 1/(1 - x) = [ (1 - x) - 1 ] / (1 - x) = (-x)/(1 - x) $. Yes, so denominator is $ -x/(1 - x) $, so reciprocal is $ - (1 - x)/x = (x - 1)/x $. So that seems correct. Therefore, $ a_3 = (x - 1)/x $. Then setting equal to $ a_1 = x $, so:

$ (x - 1)/x = x $

Multiply both sides by x:

$ x - 1 = x^2 $

Which is $ x^2 - x + 1 = 0 $, discriminant negative. So complex roots?

But maybe the problem is okay with complex numbers? But the question says "compute $ (a_9)^9 $", but doesn't specify. However, maybe I made an error in assuming $ a_3 = a_1 $, so perhaps there's another way. Wait, maybe I need to check if there is a periodicity?

Wait, if $ a_3 = a_1 $, then maybe the sequence is periodic with period 2? So if $ a_1 = a_3 $, then $ a_2 = a_4 $, etc. Then, if the sequence is periodic with period 2, then $ a_9 $ would be equal to $ a_1 $, because 9 is odd. Wait, but 9 divided by 2 is 4 with remainder 1. So if the period is 2, then $ a_9 = a_1 $, so $ a_9^9 = (a_1)^9 $. But then we need to compute $ a_1 $, but since the quadratic equation has complex roots, maybe there is some specific value?

Wait, but maybe there is a mistake here. Maybe the problem is intended for real numbers? If there are no real solutions, then maybe the problem is ill-posed? But that seems unlikely. Maybe I made an error in the calculation?

Alternatively, maybe I need to consider that the sequence could have a longer period? Let's see. Suppose that $ a_3 = a_1 $, but maybe there's another relation?

Alternatively, perhaps there's a miscalculation here. Let me check again.

Let me try plugging in some numbers. Let me suppose that $ a_1 $ is such that $ a_3 = a_1 $.

Alternatively, perhaps I need to consider that applying the recurrence three times brings us back to the original value? Wait, but the problem says $ a_3 = a_1 $. So if we start with $ a_1 $, then $ a_2 = 1/(1 - a_1) $, $ a_3 = 1/(1 - a_2) $, and then $ a_3 = a_1 $. So the function mapping $ a_1 \to a_3 $ is the same as the function composed twice. So maybe the function f(x) = 1/(1 - x), then f(f(x)) = ?

Let me compute f(f(x)):

f(x) = 1/(1 - x)

f(f(x)) = 1 / [1 - f(x)] = 1 / [1 - 1/(1 - x)] as before, which simplifies to (x - 1)/x, which is what we had earlier. So f(f(x)) = (x - 1)/x.

So setting f(f(x)) = x gives (x - 1)/x = x, leading to x^2 - x + 1 = 0. Which is complex. So maybe complex numbers are allowed here?

But then, maybe even though the roots are complex, we can still compute $ a_9^9 $. Wait, but perhaps there's a different approach. Let me consider the sequence.

Since $ a_3 = a_1 $, then the sequence is periodic with period 2? Wait, not necessarily. Because if $ a_1 = a_3 $, then $ a_2 = a_4 $, $ a_3 = a_5 $, etc. So the period is 2. So then, since 9 is odd, $ a_9 = a_1 $. Therefore, $ a_9^9 = (a_1)^9 $. Therefore, if we can find $ a_1 $, then we can compute this value.

But since $ a_1 $ satisfies $ x^2 - x + 1 = 0 $, then $ x^2 = x - 1 $. Maybe we can use this to compute $ x^9 $?

Alternatively, perhaps we can find a pattern in the powers of x. Let me try that.

Given that $ x^2 = x - 1 $, then we can express higher powers of x in terms of lower powers. Let me compute:

Let me note that:

x^2 = x - 1

x^3 = x * x^2 = x(x - 1) = x^2 - x = (x - 1) - x = -1

Wait, really? Let me compute step by step:

x^2 = x - 1.

x^3 = x * x^2 = x*(x - 1) = x^2 - x. But x^2 is x - 1, so substitute:

x^3 = (x - 1) - x = -1. So x^3 = -1.

Interesting! Then x^3 = -1.

Then x^4 = x * x^3 = x*(-1) = -x

x^5 = x * x^4 = x*(-x) = -x^2 = -(x - 1) = -x + 1

x^6 = x * x^5 = x*(-x + 1) = -x^2 + x = -(x - 1) + x = -x + 1 + x = 1

Wait, x^6 = 1?

Then x^6 = 1. So x^6 = 1. Then x^9 = x^(6 + 3) = x^6 * x^3 = 1 * (-1) = -1. Therefore, x^9 = -1.

Therefore, if a1 = x, then a9 = x, so (a9)^9 = x^9 = -1. Therefore, the answer is -1?

But let me check this again. Let me retrace:

Starting from x^2 = x - 1.

x^3 = x * x^2 = x(x - 1) = x^2 - x = (x - 1) - x = -1. Correct.

x^3 = -1.

x^4 = x * x^3 = x*(-1) = -x.

x^5 = x * x^4 = x*(-x) = -x^2 = -(x - 1) = -x + 1.

x^6 = x * x^5 = x*(-x + 1) = -x^2 + x = -(x - 1) + x = -x + 1 + x = 1. Correct. So x^6 = 1.

Therefore x^6 = 1, so x^9 = x^6 * x^3 = 1 * (-1) = -1. Therefore, yes, (a_9)^9 = -1. So the answer is \boxed{-1}? Wait, but let me check again if this is consistent.

But let me also check if there are other possibilities. Since the roots of x^2 - x + 1 = 0 are complex numbers. Let me find them. The roots are [1 ± sqrt(-3)] / 2 = [1 ± i√3]/2. These are complex numbers on the unit circle. In fact, they are the primitive sixth roots of unity? Because x^6 = 1, so they are sixth roots of unity. Let me check. Let me compute x^6 for x = [1 + i√3]/2.

Wait, [1 + i√3]/2 is e^(iπ/3), right? Because cos(π/3) = 1/2, sin(π/3) = √3/2. Therefore, raising this to the 6th power: [e^(iπ/3)]^6 = e^(i2π) = 1. Similarly, [1 - i√3]/2 is e^(-iπ/3), and raising it to the 6th power gives 1 as well. So yes, x^6 = 1. Therefore, x is a primitive 6th root of unity? Well, since x^3 = -1, which is e^(iπ), so x^3 = -1, so x^6 = 1. Therefore, x is a primitive 6th root of unity. Therefore, x^6 = 1, so x^9 = x^(6 + 3) = x^6 * x^3 = 1 * (-1) = -1. So that's consistent. Therefore, regardless of which root we take, x^9 = -1. Therefore, (a_9)^9 = -1.

Therefore, the answer is \boxed{-1}?

But let me check once again if my assumption that the sequence is periodic with period 2 is correct. Given that a3 = a1, so a1 = a3, then a4 = 1/(1 - a3) = 1/(1 - a1) = a2, so a4 = a2. Then a5 = 1/(1 - a4) = 1/(1 - a2) = a3 = a1. Similarly, a6 = a4 = a2, a7 = a5 = a1, and so on. So indeed, the sequence is periodic with period 2. Therefore, odd terms are equal to a1, even terms equal to a2. Therefore, since 9 is odd, a9 = a1. Therefore, (a9)^9 = (a1)^9 = -1. So the answer is -1.

But let me check with actual numbers. Suppose I pick a complex number x such that x^2 - x + 1 = 0. Let me take x = [1 + i√3]/2. Then compute x^3, x^6, x^9.

First, x = e^(iπ/3). Then x^3 = e^(iπ) = -1. Then x^6 = e^(i2π) = 1. Then x^9 = e^(i3π) = e^(iπ) = -1. So indeed, x^9 = -1. Similarly for the other root, x = e^(-iπ/3), same result. Therefore, regardless of the root, x^9 is -1. Therefore, (a_9)^9 = -1.

Therefore, I think that the answer is \boxed{-1}. But wait, let me check if there's any mistake in assuming that a9 = a1. Since the sequence has period 2, then:

a1, a2, a1, a2, a1, a2, a1, a2, a1,...

So a9 is the 9th term. Since the first term is a1, then term 1: a1, term 2: a2, term 3: a1, term 4: a2, term 5: a1, term 6: a2, term 7: a1, term 8: a2, term 9: a1. Yes, so a9 = a1. Therefore, (a9)^9 = (a1)^9 = -1. Therefore, the answer is indeed -1.

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

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

We are given a recurrence relation for a sequence $(a_n)$ defined by:

$$
a_{i+1} = \frac{1}{1 - a_i}
$$

with the condition $a_3 = a_1$. The goal is to compute $(a_9)^9$.

---

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

Let $a_1 = x$. Then:

- $a_2 = \frac{1}{1 - x}$
- $a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - \frac{1}{1 - x}} = \frac{1}{\left( \frac{(1 - x) - 1}{1 - x} \right)} = \frac{1 - x}{-x} = \frac{x - 1}{x}$

We are told that $a_3 = a_1$. Therefore:

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

Multiply both sides by $x$ (assuming $x \ne 0$):

$$
x - 1 = x^2 \Rightarrow x^2 - x + 1 = 0
$$

---

### Step 2: Analyze the quadratic equation

The equation $x^2 - x + 1 = 0$ has a discriminant:

$$
D = (-1)^2 - 4(1)(1) = 1 - 4 = -3
$$

So the solutions are:

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

These are complex conjugate roots, and they are the **primitive 6th roots of unity** (since they satisfy $x^6 = 1$).

---

### Step 3: Determine the periodicity of the sequence

We observe the recurrence:

$$
a_{i+1} = \frac{1}{1 - a_i}
$$

Let’s define the function $f(x) = \frac{1}{1 - x}$. Then the sequence evolves as:

$$
a_1 = x \\
a_2 = f(x) \\
a_3 = f(f(x)) \\
a_4 = f(f(f(x))) = f(a_3) \\
\text{and so on}
$$

Given $a_3 = a_1$, we can conclude that the sequence is periodic with **period 2**:

$$
a_1 = a_3 = a_5 = a_7 = a_9 = \dots \\
a_2 = a_4 = a_6 = a_8 = \dots
$$

So, since 9 is odd, $a_9 = a_1 = x$.

---

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

Since $a_9 = a_1 = x$, we compute:

$$
(a_9)^9 = x^9
$$

We already established that the roots of $x^2 - x + 1 = 0$ are the primitive 6th roots of unity. Therefore, $x^6 = 1$. Thus:

$$
x^9 = x^{6 + 3} = x^6 \cdot x^3 = 1 \cdot x^3
$$

From earlier, we saw that:

$$
x^3 = \frac{1}{1 - \frac{1}{1 - x}} = \frac{x - 1}{x}
$$

But from the equation $x^2 = x - 1$, we can compute:

$$
x^3 = x \cdot x^2 = x(x - 1) = x^2 - x = (x - 1) - x = -1
$$

Therefore:

$$
x^9 = -1
$$

---

### ✅ 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 $a_1^9 = (-1)^3 = \boxed{-1}.$