Explanation:
Okay, so I need to find the measure of angle BAC, where there's a square and a regular heptagon sharing a common side AD. Let me try to visualize this. The Asymptote code draws a regular heptagon and then defines points A, F, C, D, and B. From the labels, A is a vertex of both the square and the heptagon, and D is another common vertex. The square has vertices A, C, D, and another point, probably F? Wait, the Asymptote code defines C as A + dir(90)*(F - A). Hmm, maybe I need to think more carefully.

First, let me recall that a regular heptagon has all sides equal and all internal angles equal. The measure of each internal angle in a regular heptagon can be calculated using the formula ((n-2)*180)/n, where n is 7. So that would be (5*180)/7 ≈ 128.57 degrees. But maybe I need exact values.

Also, the square has internal angles of 90 degrees. Since they share a common side AD, points A and D are common to both the square and the heptagon. Let me try to sketch mentally: the square has sides AD, DC, CB, and BA? Wait, but the Asymptote code mentions points A, C, D, F, and B. Let me try to parse the Asymptote code.

The Asymptote code first draws a regular heptagon with vertices at dir(360*i/7 + 90) for i from 0 to 7. The dir function in Asymptote gives points on the unit circle at angles specified. So starting from 90 degrees, each subsequent vertex is 360/7 degrees apart. So the heptagon is centered at the origin, and the first vertex is at 90 degrees (top), then each next one is 360/7 degrees clockwise.

Then, point A is defined as dir(360*3/7 + 90). Let me compute that angle: 360*(3/7) is approximately 154.2857 degrees, plus 90 is 244.2857 degrees. So point A is located at 244.2857 degrees on the unit circle. Similarly, point F is dir(360*4/7 + 90). 360*(4/7) is about 205.714 degrees, plus 90 is 295.714 degrees. So point F is at 295.714 degrees.

Then point C is defined as A + dir(90)*(F - A). Hmm, dir(90) is the unit vector pointing upwards, which is (0,1). So dir(90)*(F - A) is a vector pointing upwards with magnitude equal to the length of F - A. Then adding that to point A gives point C. So this seems like moving from point A in the upward direction (perpendicular to the horizontal?) by the length of F - A? Wait, maybe not. Wait, if you take the vector from A to F, which is F - A, and then scale it by dir(90), which is (0,1), but actually, dir(90) is a unit vector. Wait, no, the Asymptote code is using dir(90) as a direction, so multiplying by (F - A) would be scaling the direction vector by the length of F - A? Wait, no, in Asymptote, if you have a vector, say, v, then dir(theta) is a unit vector. So dir(90)*(F - A) would be scaling the vector (F - A) by the direction of 90 degrees? Wait, no, actually, in Asymptote, if you have a vector, say, (x,y), then dir(theta) is a unit vector, so multiplying them would be a vector in the direction of theta with magnitude equal to the magnitude of (F - A). Wait, no, actually, in Asymptote, if you have a point P and you do P + dir(theta)*length, that would move P in the direction theta by length. But here, it's dir(90)*(F - A). Wait, maybe the code is trying to construct point C by moving from A in the direction of 90 degrees (upwards) by the length of the vector from A to F? Wait, but F is a point on the heptagon. Let me think.

Alternatively, maybe the square has side AD, so AD is a side of both the square and the heptagon. Then, since AD is a side of the square, the square has sides AD, DC, CB, and BA. But in the Asymptote code, point C is defined as A + dir(90)*(F - A). Let me try to figure out coordinates.

Let me assign coordinates. Let me assume the heptagon is centered at the origin, with vertices on the unit circle. Let me compute coordinates for points A and F.

Point A is at angle 360*(3/7) + 90 degrees. Let me convert that to radians? Wait, maybe not. Let me compute the coordinates. The dir(theta) function in Asymptote gives (cos(theta), sin(theta)), where theta is in degrees. So point A is at angle 90 + 360*(3/7) degrees. Let me compute that angle:

First, 360*(3/7) = 1080/7 ≈ 154.2857 degrees. Adding 90 degrees gives 244.2857 degrees. So coordinates of A are (cos(244.2857°), sin(244.2857°)). Similarly, point F is at angle 90 + 360*(4/7) = 90 + 1600/7 ≈ 90 + 228.571 ≈ 318.571 degrees. Coordinates of F are (cos(318.571°), sin(318.571°)).

But maybe instead of calculating exact coordinates, I can use complex numbers or vectors. Alternatively, since the heptagon is regular, the central angles between adjacent vertices are 360/7 degrees. So the angle between each adjacent vertex from the center is 360/7 degrees.

But maybe I need to figure out the positions of points A, B, D, C. Let me note that in the Asymptote code, point D is defined as C + F - A. Wait, the code says:

pair D = C + F - A;

So D = C + (F - A). Since C is A + dir(90)*(F - A), then D = [A + dir(90)*(F - A)] + (F - A) = A + dir(90)*(F - A) + F - A = dir(90)*(F - A) + F. So D = F + dir(90)*(F - A).

Alternatively, maybe there's a better way. Since AD is a common side of the square and the heptagon, so AD is a side of both. Therefore, in the heptagon, AD is a side, so points A and D are adjacent vertices of the heptagon. Wait, but in the Asymptote code, point D is defined as C + F - A. Wait, maybe I need to check the positions.

Alternatively, maybe I can think of the square and the heptagon sharing side AD. So AD is a side of the square and a side of the heptagon. Therefore, in the heptagon, moving from A to D is one side, and in the square, moving from A to D is also a side. Then, the square has another side from D to C, and from C to B, and from B to A? Wait, but the Asymptote code draws A--C--D--F. So maybe the square is A-C-D-F? Wait, but that would make a square? If A-C-D-F is a square, then AC and CD and DF and FA would be sides? But since AD is a side of the square, then maybe A-C-D is part of the square? Wait, this is getting confusing. Maybe I need to think step by step.

First, since AD is a common side, let me consider that in the heptagon, AD is a side. So in the regular heptagon, each side corresponds to a central angle of 360/7 degrees. Therefore, the angle between two adjacent vertices from the center is 360/7 degrees.

Now, the square has side AD. So the square has sides of length equal to the side length of the heptagon. Let me assume that the heptagon is regular with side length s, and the square also has side length s.

But since the square is constructed on the side AD, which is also a side of the heptagon, then the square is attached to the heptagon along AD. The Asymptote code draws A--C--D--F. So from A to C to D to F. If AD is a side of the square, then maybe AC and CD are sides of the square? Wait, but if AD is a side of the square, then the square would have sides AD, DC, CB, and BA. But in the Asymptote code, they draw A--C--D--F. So maybe C is adjacent to A and D? If AD is a side of the square, then if C is adjacent to A, then AC would be another side of the square. Similarly, CD would be another side. But then D is connected to F? Maybe F is another vertex of the square? But the Asymptote code also mentions point B, which is dir(360*2/7 + 90). Let me check that angle. 360*(2/7) is about 102.857 degrees, plus 90 is 192.857 degrees. So point B is at 192.857 degrees on the unit circle.

So maybe the square has vertices A, B, C, D? But then the Asymptote code draws A--C--D--F. Maybe F is another point? Wait, the Asymptote code draws the heptagon first, then defines points A, F, C, D, B. Then draws A--C--D--F. So maybe the square is A-C-D-F? But then A-C-D-F would be a square? Let me check. If AD is a side of the square, then AC and DF would be other sides? Wait, but if AD is a side, then the square would have sides AD, DC, CF, and FA? But then the Asymptote code draws A--C--D--F. So A to C, C to D, D to F, and then presumably F back to A? If that's the case, then A-C-D-F-A would be the square. So that would make sense. Then, since AD is a side of the square, then AC and DF would be sides of the square? Wait, but if AD is a side, then AC would be adjacent to AD? Wait, maybe not. Let me think.

Alternatively, maybe the square is A-B-C-D? But then the Asymptote code draws A--C--D--F. Maybe the square is A-C-D-F? If that's the case, then AC, CD, DF, and FA are sides of the square. But then FA would be a side. However, point F is a vertex of the heptagon. So FA would be a side of the square and also a side of the heptagon? Wait, but in the heptagon, adjacent vertices are connected by sides. If F is adjacent to A in the heptagon, then FA would be a side of the heptagon. But if FA is also a side of the square, then the square and heptagon share two sides? But the problem says they share a common side AD. So maybe FA is not a side of the square.

This is getting too confusing. Maybe I need to approach this with coordinates. Let me try to assign coordinates to the points.

Assume the heptagon is centered at the origin, with vertices on the unit circle. Let me compute coordinates for points A, D, F, B.

First, point A is at angle 90 + 360*(3/7) degrees. Let me compute that angle:

3/7 of 360 is 360*(3/7) ≈ 154.2857 degrees. Adding 90 degrees gives 244.2857 degrees. So point A is at 244.2857 degrees. Similarly, point D is... Wait, the problem says they share side AD. So if AD is a side of the heptagon, then D is the next vertex after A in the heptagon. Since the heptagon is drawn with vertices at angles starting from 90 degrees, increasing by 360/7 degrees each time. So if point A is at angle θ, then the next vertex (D) would be at θ + 360/7 degrees. Wait, but in the Asymptote code, point A is defined as dir(360*3/7 + 90). Let me check the indices. The Asymptote code draws the heptagon with vertices from i=0 to 7, so 8 points? Wait, no, for(int i=0; i <=7; ++i) would be 8 points, but a heptagon has 7 sides. Wait, maybe it's a typo? Or maybe it's drawing a heptagon with 7 sides, so i from 0 to 6? Maybe the Asymptote code has a mistake? But regardless, the code defines point A as dir(360*3/7 + 90). Let me think of the heptagon as having vertices labeled 0 through 6, each separated by 360/7 degrees. So starting from angle 90 degrees (which would be vertex 0), then vertex 1 is at 90 + 360/7 degrees, vertex 2 at 90 + 2*(360/7), etc. So point A is vertex 3: 90 + 3*(360/7) degrees. Then, the next vertex (vertex 4) would be at 90 + 4*(360/7) degrees, which is point F. Therefore, AD is the side from vertex 3 to vertex 4, which is the side between A and F? Wait, but in the Asymptote code, point D is defined as C + F - A. Wait, maybe there's confusion here.

Alternatively, maybe the Asymptote code defines point D as the next vertex after A in the heptagon? If A is vertex 3, then the next vertex is vertex 4 (point F). But then AD would be AF? But the problem says they share side AD. So maybe D is vertex 3 + 1? Wait, maybe I need to check the labels. The Asymptote code labels A, B, C, D, F. So maybe D is a different point?

Alternatively, maybe the square is constructed externally on the side AD of the heptagon. So AD is a side of both the heptagon and the square. Then, the square extends from AD in some direction. Since the heptagon is regular, the side AD is part of the heptagon, and the square is built on AD.

To find angle BAC, we need to figure out the positions of points A, B, and C.

Let me try to think of the square. If AD is a side of the square, then the square has sides AD, DC, CB, and BA. So from point A, moving along AD to D, then from D moving perpendicularly to AD to get to C, then from C moving back to B, and from B back to A. But since the square is in the plane with the heptagon, we need to figure out the direction of the square relative to the heptagon.

Alternatively, since the Asymptote code defines point C as A + dir(90)*(F - A). Let me think of this as moving from point A in the direction of 90 degrees (upwards) by the vector from A to F. Wait, if F is another point on the heptagon, then vector AF is from A to F. If we take that vector and scale it in the direction of 90 degrees, then add it to A, we get point C.

Alternatively, maybe dir(90) is a unit vector, so dir(90)*(F - A) is scaling the vector (F - A) by the direction of 90 degrees? Wait, no, in Asymptote, if you have a vector v, then dir(theta) * v would be a vector in the direction theta with magnitude equal to the magnitude of v? Wait, no, actually, dir(theta) is a unit vector, so multiplying by a vector would be a bit ambiguous. Wait, maybe the code is using vector operations. If you have a point A and a vector, then adding the vector to A gives another point. So dir(90) is a unit vector pointing upwards. So dir(90)*(F - A) would be scaling the vector (F - A) by the direction of 90 degrees? Wait, no, that doesn't make sense. Maybe it's a typo or misunderstanding.

Alternatively, maybe the code is trying to construct point C by moving from A in the direction perpendicular to AD? Since AD is a side of the square, then the square would have sides perpendicular to AD. If AD is a side of the heptagon, then the direction of AD can be determined.

Let me try to compute coordinates. Let me assume the heptagon is centered at the origin, with radius 1. Let me compute coordinates for points A and D (assuming D is the next vertex after A).

Point A is at angle θ = 90 + 3*(360/7) degrees. Let me compute θ:

First, 360/7 ≈ 51.4286 degrees. So 3*(360/7) ≈ 154.2857 degrees. Adding 90 degrees gives θ ≈ 244.2857 degrees. So coordinates of A are (cos(244.2857°), sin(244.2857°)). Similarly, the next vertex D would be at θ + 360/7 ≈ 244.2857 + 51.4286 ≈ 295.7143 degrees. So coordinates of D are (cos(295.7143°), sin(295.7143°)).

Now, vector AD is D - A. Let me compute that vector. But since both points are on the unit circle, the vector AD can be calculated. However, since we need to construct the square on AD, we need to find the direction perpendicular to AD.

In the square, the side AD is one side, and the adjacent sides are perpendicular to AD. So if we can find the direction of the square relative to the heptagon, we can find points B and C.

Alternatively, since the Asymptote code defines point C as A + dir(90)*(F - A), and point F is the next vertex after A in the heptagon. Wait, earlier I thought point F was the next vertex after A, which would be D? Wait, no, if point A is vertex 3, then the next vertex is vertex 4, which is F. Then D is another point? Wait, the Asymptote code defines D as C + F - A. So if I can figure out coordinates for C, then I can find D.

Alternatively, maybe I need to use complex numbers for easier calculations. Let me represent points as complex numbers. Let me denote the center of the heptagon as the origin.

Let me denote point A as a complex number: A = e^{iθ}, where θ = 90° + 3*(360°/7). Similarly, point F is the next vertex, so F = e^{i(θ + 360°/7)}.

Then, vector AF is F - A. The Asymptote code defines point C as A + dir(90)*(F - A). dir(90) is e^{i90°} = i. So multiplying by i rotates the vector AF by 90 degrees counterclockwise. Therefore, point C = A + i*(F - A).

Let me compute that.

C = A + i*(F - A) = A - i*A + i*F = A*(1 - i) + i*F.

Then, point D is defined as C + F - A. Let me compute that:

D = C + F - A = [A*(1 - i) + i*F] + F - A = A*(1 - i) - A + i*F + F = A*(1 - i - 1) + F*(i + 1) = A*(-i) + F*(1 + i).

So D = -i*A + (1 + i)*F.

But since A and F are points on the unit circle, maybe we can compute coordinates. However, this might get complicated. Alternatively, maybe there's a geometric interpretation.

Since AD is a side of the square, then the square has sides AD, DC, CB, and BA. If we can find coordinates of C and B, then we can compute angle BAC.

Alternatively, since we have point C defined as A + i*(F - A), which is a rotation of vector AF by 90 degrees counterclockwise and adding it to A. So if we imagine vector AF, rotating it 90 degrees counterclockwise gives a vector perpendicular to AF. Adding that to A gives point C.

But since AD is a side of the square, and AF is a side of the heptagon, which is adjacent to AD? Wait, if A is connected to D and to F in the heptagon, then AF is a diagonal? Wait, no, in a regular heptagon, each vertex is connected to its adjacent vertices. So if A is connected to D, then D is the next vertex after A, so F would be the vertex after D? Wait, no, if A is vertex 3, then D is vertex 4 (F?), and then the next vertex is vertex 5? Wait, this is confusing.

Wait, in the Asymptote code, point F is defined as dir(360*4/7 + 90). So if the first vertex is at i=0: 90 degrees, then i=1: 90 + 360/7, i=2: 90 + 2*360/7, i=3: 90 + 3*360/7 (which is point A), i=4: 90 + 4*360/7 (which is point F). Therefore, in the heptagon, the vertices are labeled 0 to 6, with A being vertex 3, F being vertex 4, and so on. Therefore, the side AD would be from vertex 3 to vertex 4 (A to F)? But the problem says they share side AD. So maybe D is vertex 4? But in the Asymptote code, point D is defined as C + F - A. So maybe D is not a vertex of the heptagon? Wait, but the problem says they share a common side AD, so AD must be a side of both the square and the heptagon. Therefore, AD is a side of the heptagon, meaning that D is the next vertex after A in the heptagon. Therefore, if A is vertex 3, then D is vertex 4 (which is point F). But then the Asymptote code defines point F as dir(360*4/7 + 90), which would be vertex 4. Therefore, maybe there's a typo in the Asymptote code, and point D is actually point F? Or maybe the label D is different?

Alternatively, maybe the Asymptote code has a mistake, and point D is supposed to be another point. Given that the problem says they share side AD, and in the Asymptote code, point D is defined as C + F - A, which is a different point.

This is getting too tangled. Maybe I need to approach this with angles.

Let me recall that in a regular heptagon, the central angle between two adjacent vertices is 360/7 degrees. The internal angle at each vertex is ((7-2)/7)*180 = 128.571 degrees.

Now, angle BAC is the angle at point A between points B and C. To find this angle, I need to know the positions of points B, A, and C relative to each other.

Assuming that AD is a side of both the square and the heptagon, then in the heptagon, the side AD is between vertices A and D. In the square, AD is a side, so the square extends from A to D, and then from D, the square goes in a direction perpendicular to AD.

Let me consider the square. If AD is a side of the square, then the square has sides AD, DC, CB, and BA. So from point A, moving to D, then turning 90 degrees to get to C, then to B, then back to A.

To find angle BAC, which is the angle between BA and CA.

But to find this angle, I need to know the positions of B and C relative to A.

Alternatively, since the square is constructed on AD, and the heptagon is regular, maybe we can find the angles between the sides.

Let me consider the heptagon first. Let me denote the center of the heptagon as O. The central angle between A and D is 360/7 degrees, since they are adjacent vertices.

Now, in the square, AD is a side, and the square is constructed such that from A, moving to D, then turning 90 degrees to get to C. The direction of this turn depends on whether the square is inside or outside the heptagon.

Assuming the square is constructed externally on the side AD, then the direction from D to C would be perpendicular to AD.

To find the angle BAC, we need to relate the positions of B and C with respect to A.

Alternatively, maybe using complex numbers would help. Let me assign complex numbers to points. Let me assume the heptagon is centered at the origin, with radius 1. Let me denote point A as a complex number, and point D as another complex number.

Let me denote the angle for point A as θ_A = 90° + 3*(360°/7). Similarly, point D is θ_D = θ_A + 360°/7.

The vector AD is D - A. In the square, the next side from D is DC, which is perpendicular to AD. So the direction from D to C is perpendicular to AD.

If we consider the square, then from D, moving in a direction perpendicular to AD to reach C. Similarly, from A, moving in a direction perpendicular to AD to reach B? Wait, no. In the square, from A, moving along AD to D, then from D, moving perpendicularly to AD to get to C, then from C, moving back towards B, and from B back to A.

Alternatively, since AD is a side of the square, then AB and AD are adjacent sides of the square, meaning that AB is perpendicular to AD. Wait, but if AD is a side of the square, then AB would be another side, so AB is perpendicular to AD. But in the problem, point B is a vertex of the heptagon. Wait, the Asymptote code defines point B as dir(360*2/7 + 90). Let me compute that angle: 360*(2/7) ≈ 102.857°, plus 90° is 192.857°, which is in the third quadrant.

So point B is located at 192.857°, which is between 180° and 270°, so southwest direction.

If I can figure out the relationship between points A, B, and C, maybe I can compute the angle.

Alternatively, since angle BAC is formed by points B, A, and C, and we need to find its measure.

Let me think about the positions. If AD is a side of the square, then AB is another side of the square? Or is AB a diagonal?

Wait, if AD is a side of the square, then AB could be another side if the square is constructed such that A is connected to B and D. But since the square has four sides, if AD is one side, then AB would be another side, making angle BAD 90 degrees.

But point B is a vertex of the heptagon. So if AB is a side of the square and also a side of the heptagon, then AB would be a side of both. But the problem says they share only side AD. Therefore, AB is not a side of the heptagon. Therefore, AB is a side of the square but not of the heptagon.

Therefore, the square has sides AD, DC, CB, and BA, where BA is not a side of the heptagon.

Therefore, to find angle BAC, which is the angle at A between BA and CA.

To find this angle, I need to find the angle between vectors AB and AC.

Since AB is a side of the square, and AC is a diagonal of the square? Wait, no. If AD is a side of the square, and AB is another side, then AC would be the diagonal from A to C, which would be the diagonal of the square. But in that case, angle BAC would be 45 degrees, but that seems too simple, and the answer is supposed to be a common fraction, which might not be 45 degrees.

Wait, but maybe not. Let me think again. If AD is a side of the square, and AB is another side, then AB is perpendicular to AD. So angle BAD is 90 degrees. Then, if AC is the diagonal from A to C, then angle BAC would be half of 90 degrees, which is 45 degrees. But this seems too straightforward, and the problem involves a heptagon, so maybe there's more to it.

Alternatively, maybe AC is not the diagonal of the square. If the square is A-D-C-B-A, then AC would be a diagonal, yes. But if AB is a side, then angle BAC would be 45 degrees. But since the heptagon is involved, maybe the position of point C is different?

Wait, but according to the Asymptote code, point C is defined as A + dir(90)*(F - A). If F is the next vertex after A in the heptagon, then vector AF is a side of the heptagon. Rotating that vector by 90 degrees and adding to A gives point C.

If I consider that, then AC is a vector obtained by rotating AF by 90 degrees. So the angle between AF and AC is 90 degrees.

But AF is a side of the heptagon, and AC is a side of the square? Or is it?

Alternatively, since AD is a side of the square, and AF is a side of the heptagon, which is adjacent to AD? Wait, if A is connected to D and to F in the heptagon, then AF is a side of the heptagon, and AD is also a side. But in a regular heptagon, each vertex is connected to two adjacent vertices. So if AD is a side, then A is connected to D and to another vertex, say, let's call it E. Wait, but in the Asymptote code, point B is defined as dir(360*2/7 + 90). Let me check the positions again.

If the heptagon has vertices labeled 0 to 6, with vertex 0 at 90°, then vertex 1 is at 90° + 360/7°, vertex 2 at 90° + 2*360/7°, vertex 3 at 90° + 3*360/7° (which is point A), vertex 4 at 90° + 4*360/7° (point F), vertex 5 at 90° + 5*360/7°, vertex 6 at 90° + 6*360/7°, and back to vertex 0.

So, vertex A is vertex 3, connected to vertex 2 and vertex 4. Therefore, in the heptagon, point A is connected to vertex 2 (which would be point B?) and vertex 4 (which is point F). Wait, the Asymptote code defines point B as dir(360*2/7 + 90), which is vertex 2. So yes, point B is vertex 2, connected to A (vertex 3) and vertex 1.

Therefore, in the heptagon, side AB is between vertex 2 and 3, which is a side of the heptagon. But the problem says that the square and heptagon share side AD. So AD is a side of both, meaning that AD is a side of the heptagon (between vertex 3 and 4) and also a side of the square.

Therefore, the square has side AD, and since AB is a side of the heptagon, it is not a side of the square. Therefore, the square is constructed on side AD, extending outward from the heptagon.

Now, to find angle BAC, which is the angle at A between points B (vertex 2 of the heptagon) and C (a point defined in the Asymptote code).

To find this angle, I need to determine the positions of points B and C relative to A.

Since point B is vertex 2 of the heptagon, and point A is vertex 3, the angle between AB and the center is known. Similarly, point C is defined as A + dir(90)*(F - A). Let me try to compute this.

Let me consider complex numbers again. Let me denote the center of the heptagon as the origin, and all points on the unit circle.

Let me denote:

- Point A: e^{iθ}, where θ = 90° + 3*(360°/7)
- Point F: e^{i(θ + 360°/7)} (next vertex)
- Point B: e^{i(θ - 360°/7)} (previous vertex, since A is vertex 3, B is vertex 2)

Wait, if θ is the angle for vertex 3, then vertex 2 is θ - 360°/7, and vertex 4 is θ + 360°/7.

So, vector AF is F - A = e^{i(θ + 360°/7)} - e^{iθ} = e^{iθ}(e^{i360°/7} - 1).

Then, dir(90°) is e^{i90°} = i. So point C = A + i*(F - A) = e^{iθ} + i*(e^{iθ}(e^{i360°/7} - 1)) = e^{iθ}[1 + i(e^{i360°/7} - 1)].

This seems complex, but maybe we can simplify. Let me compute the term inside the brackets:

1 + i(e^{iα} - 1), where α = 360°/7.

Let me write e^{iα} = cosα + i sinα. Then:

i(e^{iα} - 1) = i(cosα + i sinα - 1) = i(cosα - 1) + i^2 sinα = i(cosα - 1) - sinα.

Therefore, the entire expression becomes:

1 - sinα + i(cosα - 1).

So point C = e^{iθ} * [1 - sinα + i(cosα - 1)].

This is getting complicated, but maybe we can find the angle of point C relative to A.

Alternatively, since we need angle BAC, which is the angle between vectors AB and AC.

Vector AB is B - A = e^{i(θ - α)} - e^{iθ} = e^{iθ}(e^{-iα} - 1).

Vector AC is C - A = [A + i(F - A)] - A = i(F - A) = i(e^{iθ}(e^{iα} - 1)) = e^{iθ} * i(e^{iα} - 1).

So vector AC = e^{iθ} * i(e^{iα} - 1).

Therefore, the angle between AB and AC can be found by taking the argument of (AC / AB).

Let me compute AC / AB:

AC / AB = [e^{iθ} * i(e^{iα} - 1)] / [e^{iθ}(e^{-iα} - 1)] = [i(e^{iα} - 1)] / [e^{-iα} - 1].

Simplify denominator: e^{-iα} - 1 = - (1 - e^{-iα}) = -e^{-iα/2} * 2i sin(α/2) [using identity 1 - e^{-iα} = 2i sin(α/2) e^{-iα/2}]. Wait, let me recall that 1 - e^{-iα} = e^{-iα/2} * 2i sin(α/2).

Wait, more carefully:

Let me write 1 - e^{-iα} = e^{-iα/2} * [e^{iα/2} - e^{-iα/2}] = e^{-iα/2} * 2i sin(α/2).

Therefore, e^{-iα} - 1 = - (1 - e^{-iα}) = -e^{-iα/2} * 2i sin(α/2).

Similarly, numerator: i(e^{iα} - 1) = i * [e^{iα} - 1] = i * e^{iα/2} * [e^{iα/2} - e^{-iα/2}] = i * e^{iα/2} * 2i sin(α/2) = -2 sin(α/2) * e^{iα/2}.

Therefore, AC / AB = [ -2 sin(α/2) * e^{iα/2} ] / [ -e^{-iα/2} * 2i sin(α/2) ] = [ -2 sin(α/2) e^{iα/2} ] / [ -2i sin(α/2) e^{-iα/2} ] = [ e^{iα/2} ] / [ i e^{-iα/2} ] = e^{iα} / i = -i e^{iα}.

Because dividing by i is multiplying by -i.

So AC / AB = -i e^{iα}.

Therefore, the argument of AC / AB is the argument of -i e^{iα}.

Let me compute this.

First, e^{iα} has argument α.

-i has argument -π/2 or 3π/2.

Therefore, the total argument is α - π/2.

Therefore, the angle between vectors AB and AC is α - π/2.

But α is 360°/7, which is approximately 51.4286°.

So α - π/2 is 51.4286° - 90° = -38.5714°, which is equivalent to 360° - 38.5714° = 321.4286°, but since angles are modulo 360°, the actual angle between AB and AC is the absolute value, which is 38.5714°, or 360/7 - 90°, but let me compute it properly.

Wait, the angle between AB and AC is the argument of AC / AB, which is α - π/2. But since angles are measured from AB to AC, we need to consider the direction.

But since we're dealing with vectors, the angle between AB and AC is the absolute value of the argument, or depending on orientation.

But let me convert α to degrees: α = 360°/7 ≈ 51.4286°.

So α - π/2 is 51.4286° - 90° = -38.5714°, which is equivalent to 360° - 38.5714° = 321.4286°, but the angle between two vectors is the smallest angle between them, so it's 38.5714°, which is 360/7 - 90° + 360°? Wait, no.

Wait, the angle between AB and AC is |α - π/2|, but since α is in degrees, let me convert π/2 to degrees: π/2 = 90°. So the angle is |α - 90°| = |360/7 - 90| degrees.

Compute 360/7 ≈ 51.4286°, so 51.4286° - 90° = -38.5714°, absolute value is 38.5714°, which is 38.5714°.

But 38.5714° is equal to 270/7°, since 270 divided by 7 is approximately 38.5714. Let me check: 7*38 = 266, 7*38.5714 ≈ 270. Yes, 270/7 = 38.5714°.

Therefore, the angle between AB and AC is 270/7 degrees.

But wait, angle BAC is this angle? Wait, the angle between AB and AC is the angle at A between AB and AC, which is exactly angle BAC.

Therefore, the measure of angle BAC is 270/7 degrees.

But let me verify this.

Wait, earlier steps said that AC / AB has an argument of α - π/2, which is 360/7 - 90°, which is negative, but the angle between vectors is the absolute value. However, depending on the orientation, it could be 90° - α. Wait, let me think again.

If the argument of AC / AB is α - π/2, then the angle from AB to AC is α - π/2. But since α is less than π/2 (since 360/7 ≈ 51.4° < 90°), this would be a negative angle, meaning that AC is obtained by rotating AB clockwise by π/2 - α. Therefore, the angle between AB and AC is π/2 - α.

Wait, this is a crucial point. If the argument is negative, it means the angle is measured clockwise. The actual angle between the vectors is the smallest angle between them, which would be π/2 - α.

So π/2 - α is 90° - 360/7° = (630/7 - 360/7)° = 270/7° ≈ 38.57°, which matches the previous result.

Therefore, angle BAC is 270/7 degrees.

But let me check if this makes sense.

In the heptagon, the internal angle at A is 128.57°, which is 5*180/7 ≈ 128.57°. If angle BAC is 270/7 ≈ 38.57°, then that seems plausible.

But let me check with another approach.

Since AD is a side of the square, and AB is a side of the heptagon. The angle between AB and AD can be calculated.

In the heptagon, the internal angle at A is 128.57°, which is the angle between AB and AD. Wait, no. In a regular heptagon, each internal angle is the angle between two adjacent sides. So at vertex A, the two adjacent sides are AB and AD. Therefore, the internal angle at A is the angle between AB and AD, which is 128.57°.

But angle BAC is the angle between AB and AC. If AC is a side of the square, then we need to relate AC to AD.

Since AD is a side of the square, and AC is another side or diagonal?

Wait, earlier we found that angle between AB and AC is 270/7°, which is approximately 38.57°, and the internal angle between AB and AD is 128.57°, so angle between AC and AD would be 128.57° - 38.57° = 90°, which makes sense because AD is a side of the square, and AC is a side of the square? Wait, no. If angle between AB and AC is 38.57°, and angle between AB and AD is 128.57°, then angle between AC and AD is 128.57° - 38.57° = 90°, which would mean that AC is perpendicular to AD, which is consistent with AC being a side of the square.

Yes! Because in the square, AD is a side, and AC would be another side if the square is constructed such that from A, moving along AB (which is a side of the heptagon) and AC (which is a side of the square). Wait, no. If AD is a side of the square, then AC would be a diagonal or another side.

Wait, if angle between AC and AD is 90°, then AC is perpendicular to AD, which would mean that AC is a side of the square. Since AD is a side of the square, then AC would be another side, making angle CAD 90°, which would mean that the square has sides AD and AC. But then AB is another side?

Wait, this is getting confusing. Let me think again.

If angle between AB and AC is 270/7°, and angle between AB and AD is 128.57°, then angle between AC and AD is 128.57° - 270/7°. Let me compute 128.57° in fractions.

Since internal angle of heptagon is 5*180/7 = 900/7 ≈ 128.57°.

So angle between AC and AD is 900/7 - 270/7 = 630/7 = 90°, which is 90 degrees.

Therefore, AC is perpendicular to AD, which makes sense because AD is a side of the square, and AC is another side of the square, making a right angle at A.

Therefore, angle BAC is 270/7 degrees, which is the answer.

But let me check if 270/7 is reducible. 270 divided by 7 is 38 and 4/7, so as a fraction, it's 270/7.

But the problem says "Express your answer as a common fraction." So 270/7 is already a common fraction.

But let me check if this is correct.

Alternatively, maybe I made a mistake in the sign. Let me verify the calculation of the angle between AB and AC.

We had AC / AB = -i e^{iα}, whose argument is α - π/2. But since we're dealing with vectors, the angle between AB and AC is the absolute value of that argument, or π - |argument|?

Wait, the argument of AC / AB is the angle from AB to AC. If it's negative, it means clockwise. The actual angle between them is the smallest angle, which would be π/2 - α, as π/2 is 90°, and α is 360/7°.

So π/2 - α = 90° - 360/7° = (630/7 - 360/7)° = 270/7°, which is approximately 38.57°, which is less than 90°, so it's the correct angle between AB and AC.

Therefore, angle BAC is 270/7 degrees.

But let me check with another approach.

Let me consider the regular heptagon and the square.

In the regular heptagon, the central angles are 360/7°. The internal angle at each vertex is 5*180/7° = 900/7°.

At vertex A, the two adjacent vertices are B and D. The angle between AB and AD is 900/7°.

Now, the square has side AD, and from point A, the square extends in a direction perpendicular to AD. Since AD is a side of the square, the square's side AC is perpendicular to AD.

Therefore, angle between AD and AC is 90°, as it's a square.

Therefore, angle between AB and AC is angle between AB and AD minus angle between AC and AD, which is 900/7° - 90° = 900/7° - 630/7° = 270/7°, which matches our previous result.

Therefore, angle BAC is 270/7 degrees.

But let me check if this is the answer.

Wait, 270 divided by 7 is approximately 38.57°, which seems reasonable.

But let me check if there's an alternative way to get this.

Alternatively, since angle BAC is the difference between the internal angle of the heptagon at A and the right angle of the square.

Yes, as above.

Therefore, the measure of angle BAC is 270/7 degrees.

But let me check if the problem says "degree measure," so 270/7 is the answer.

But let me check if 270/7 reduces or if there's a miscalculation.

Wait, 270/7 is already in simplest terms.

But let me check with actual values.

Let me compute 270 divided by 7: 7*38 = 266, 270 - 266 = 4, so 270/7 = 38 4/7 degrees.

Alternatively, maybe the answer is expected in another form?

Wait, the problem says "degree measure," so 270/7 is acceptable as a common fraction.

But let me check if there's a different approach that might give a different answer.

Alternatively, using coordinates. Let me try to compute coordinates for points A, B, and C.

Assume the heptagon is unit circle.

Let me compute coordinates for point A, B, and C.

First, compute α = 360/7 ≈ 51.4286°.

Point A is at angle θ = 90° + 3α.

Let me compute θ: 90° + 3*(360/7)° = 90° + 1080/7° = 90° + 154.2857° ≈ 244.2857°.

Coordinates of A: (cos(244.2857°), sin(244.2857°)).

Similarly, point B is at angle θ - α = 244.2857° - 51.4286° ≈ 192.8571°.

Coordinates of B: (cos(192.8571°), sin(192.8571°)).

Point C is defined as A + dir(90)*(F - A).

Point F is at angle θ + α ≈ 244.2857° + 51.4286° ≈ 295.7143°.

Coordinates of F: (cos(295.7143°), sin(295.7143°)).

Vector AF is F - A.

dir(90) is (0,1). So dir(90)*(F - A) is (0,1) * (F_x - A_x, F_y - A_y) = (0, F_y - A_y).

Wait, no. If dir(90) is a unit vector, then multiplying by the vector (F - A) would be scaling? Wait, no, in Asymptote, if you have a vector v, then dir(theta) * v would be a vector in direction theta with magnitude equal to |v|. But I think in Asymptote, the * operator for vectors is component-wise multiplication, but I might be mistaken.

Alternatively, if dir(90) is a unit vector, then dir(90)*(F - A) would be the vector (F - A) rotated by 90 degrees counterclockwise.

Yes, that makes sense. Because multiplying by i in complex numbers corresponds to rotating by 90 degrees.

So if we consider F - A as a complex number, multiplying by i rotates it 90 degrees counterclockwise.

Therefore, point C = A + i*(F - A).

Let me compute this in complex numbers.

Let me denote A as a complex number: A = e^{iθ}, F = e^{i(θ + α)}.

Then, F - A = e^{i(θ + α)} - e^{iθ} = e^{iθ}(e^{iα} - 1).

Multiplying by i: i*(F - A) = i*e^{iθ}(e^{iα} - 1).

Adding to A: C = A + i*(F - A) = e^{iθ} + i*e^{iθ}(e^{iα} - 1) = e^{iθ}[1 + i(e^{iα} - 1)].

As before.

Now, to find angle BAC, we can compute the angle between vectors AB and AC.

Vector AB = B - A = e^{i(θ - α)} - e^{iθ} = e^{iθ}(e^{-iα} - 1).

Vector AC = C - A = i*(F - A) = i*e^{iθ}(e^{iα} - 1).

So, as before, the angle between AB and AC is the argument of (AC / AB) = [i(e^{iα} - 1)] / [e^{-iα} - 1].

As previously calculated, this simplifies to -i e^{iα}, whose argument is α - π/2, leading to angle BAC = π/2 - α = 90° - 360/7° = 270/7°.

Therefore, this confirms the previous result.

Thus, the measure of angle BAC is 270/7 degrees.

But let me check if this answer makes sense with the diagram.

In the Asymptote code, the square is drawn as A--C--D--F. So from A to C to D to F. If AD is a side of the square, then AC and CD are sides of the square. Therefore, angle at A between AC and AD is 90°, as it's a square.

Since angle between AB and AD is the internal angle of the heptagon, which is 900/7°, then angle between AB and AC is 900/7° - 90° = 270/7°, which matches.

Therefore, I'm confident that the answer is 270/7 degrees.

**Final Answer**
\boxed{\dfrac{270}{7}}

To determine the degree measure of angle $ \angle BAC $, we analyze the geometric configuration involving a square and a regular heptagon that share a common side $ \overline{AD} $.

---

### Step 1: Understand the Geometry

- **Regular Heptagon**: A regular heptagon has 7 equal sides and 7 equal internal angles. The internal angle at each vertex is:
  $$
  \text{Internal angle} = \frac{(7 - 2) \cdot 180^\circ}{7} = \frac{900^\circ}{7}
  $$

- **Square**: A square has four equal sides and four right angles (90°). Since the square and the heptagon share side $ \overline{AD} $, this side is common to both.

- **Point A** is a vertex of both the square and the heptagon. The square extends from $ A $ to $ D $, and from $ A $, the square continues in a direction perpendicular to $ \overline{AD} $, forming a right angle at $ A $.

- **Point B** is a vertex of the heptagon adjacent to $ A $, and **Point C** is a vertex of the square adjacent to $ A $, forming the angle $ \angle BAC $.

---

### Step 2: Use Geometric Relationships

- The internal angle at vertex $ A $ of the heptagon is $ \frac{900^\circ}{7} $, which is the angle between the sides $ \overline{AB} $ and $ \overline{AD} $.

- Since $ \overline{AD} $ is a side of the square, and the square has a right angle at $ A $, the angle between $ \overline{AD} $ and $ \overline{AC} $ is $ 90^\circ $.

- Therefore, the angle $ \angle BAC $ is the difference between the internal angle of the heptagon at $ A $ and the right angle of the square:
  $$
  \angle BAC = \frac{900^\circ}{7} - 90^\circ = \frac{900^\circ - 630^\circ}{7} = \frac{270^\circ}{7}
  $$

---

### Final Answer

$$
\boxed{\dfrac{270}{7}}
$$

[Assistant]
Explanation:
Okay, so I need to find the measure of angle BAC, where there's a square and a regular heptagon sharing a common side AD. Let me try to visualize this. The Asymptote code draws a regular heptagon and then defines points A, F, C, D, and B. From the labels, A is a vertex of both the square and the heptagon, and D is another common vertex. The square has vertices A, C, D, and another point, probably F? Wait, the Asymptote code defines C as A + dir(90)*(F - A). Hmm, maybe I need to think more carefully.

First, let me recall that a regular heptagon has all sides equal and all internal angles equal. The measure of each internal angle in a regular heptagon can be calculated using the formula ((n-2)*180)/n, where n is 7. So that would be (5*180)/7 ≈ 128.57 degrees. But maybe I need exact values.

Also, the square has internal angles of 90 degrees. Since they share a common side AD, points A and D are common to both the square and the heptagon. Let me try to sketch mentally: the square has sides AD, DC, CB, and BA? Wait, but the Asymptote code mentions points A, C, D, F, and B. Let me try to parse the Asymptote code.

The Asymptote code first draws a regular heptagon with vertices at dir(360*i/7 + 90) for i from 0 to 7. The dir function in Asymptote gives points on the unit circle at angles specified. So starting from 90 degrees, each subsequent vertex is 360/7 degrees apart. So the heptagon is centered at the origin, and the first vertex is at 90 degrees (top), then each next one is 360/7 degrees clockwise.

Then, point A is defined as dir(360*3/7 + 90). Let me compute that angle: 360*(3/7) is approximately 154.2857 degrees, plus 90 is 244.2857 degrees. So point A is located at 244.2857 degrees on the unit circle. Similarly, point F is dir(360*4/7 + 90). 360*(4/7) is about 205.714 degrees, plus 90 is 295.714 degrees. So point F is at 295.714 degrees.

Then point C is defined as A + dir(90)*(F - A). Hmm, dir(90) is the unit vector pointing upwards, which is (0,1). So dir(90)*(F - A) is a vector pointing upwards with magnitude equal to the length of F - A. Then adding that to point A gives point C. So this seems like moving from point A in the upward direction (perpendicular to the horizontal?) by the length of F - A? Wait, maybe not. Wait, if you take the vector from A to F, which is F - A, and then scale it by dir(90), which is (0,1), but actually, dir(90) is a unit vector. Wait, no, the Asymptote code is using dir(90) as a direction, so multiplying by (F - A) would be scaling the direction vector by the length of F - A? Wait, no, in Asymptote, if you have a vector, say, v, then dir(theta) is a unit vector. So dir(90)*(F - A) would be scaling the vector (F - A) by the direction of 90 degrees? Wait, no, actually, in Asymptote, if you have a vector, say, (x,y), then dir(theta) is a unit vector, so multiplying them would be a vector in the direction of theta with magnitude equal to the magnitude of (F - A). Wait, no, actually, in Asymptote, if you have a point P and you do P + dir(theta)*length, that would move P in the direction theta by length. But here, it's dir(90)*(F - A). Wait, maybe the code is trying to construct point C by moving from A in the direction of 90 degrees (upwards) by the length of the vector from A to F? Wait, but F is a point on the heptagon. Let me think.

Alternatively, maybe the square has side AD, so AD is a side of both the square and the heptagon. Then, since AD is a side of the square, the square has sides AD, DC, CB, and BA. But in the Asymptote code, point C is defined as A + dir(90)*(F - A). Let me try to figure out coordinates.

Let me assign coordinates. Let me assume the heptagon is centered at the origin, with vertices on the unit circle. Let me compute coordinates for points A and F.

Point A is at angle 360*(3/7) + 90 degrees. Let me convert that to radians? Wait, maybe not. Let me compute the coordinates. The dir(theta) function in Asymptote gives (cos(theta), sin(theta)), where theta is in degrees. So point A is at angle 90 + 360*(3/7) degrees. Let me compute that angle:

First, 360*(3/7) = 1080/7 ≈ 154.2857 degrees. Adding 90 degrees gives 244.2857 degrees. So coordinates of A are (cos(244.2857°), sin(244.2857°)). Similarly, point F is at angle 90 + 360*(4/7) = 90 + 1600/7 ≈ 90 + 228.571 ≈ 318.571 degrees. Coordinates of F are (cos(318.571°), sin(318.571°)).

But maybe instead of calculating exact coordinates, I can use complex numbers or vectors. Alternatively, since the heptagon is regular, the central angles between adjacent vertices are 360/7 degrees. So the angle between each adjacent vertex from the center is 360/7 degrees.

But maybe I need to figure out the positions of points A, B, D, C. Let me note that in the Asymptote code, point D is defined as C + F - A. Wait, the code says:

pair D = C + F - A;

So D = C + (F - A). Since C is A + dir(90)*(F - A), then D = [A + dir(90)*(F - A)] + (F - A) = A + dir(90)*(F - A) + F - A = dir(90)*(F - A) + F. So D = F + dir(90)*(F - A).

Alternatively, maybe there's a better way. Since AD is a common side of the square and the heptagon, so AD is a side of both. Therefore, in the heptagon, AD is a side, so points A and D are adjacent vertices of the heptagon. Wait, but in the Asymptote code, point D is defined as C + F - A. Wait, maybe I need to check the positions.

Alternatively, maybe I can think of the square and the heptagon sharing side AD. So AD is a side of the square and a side of the heptagon. Therefore, in the heptagon, moving from A to D is one side, and in the square, moving from A to D is also a side. Then, the square has another side from D to C, and from C to B, and from B to A? Wait, but the Asymptote code draws A--C--D--F. So maybe the square is A-C-D-F? Wait, but that would make a square? If A-C-D-F is a square, then AC and CD and DF and FA would be sides? But since AD is a side of the square, then maybe A-C-D is part of the square? Wait, this is getting confusing. Maybe I need to think step by step.

First, since AD is a common side, let me consider that in the heptagon, AD is a side. So in the regular heptagon, each side corresponds to a central angle of 360/7 degrees. Therefore, the angle between two adjacent vertices from the center is 360/7 degrees.

Now, the square has side AD. So the square has sides of length equal to the side length of the heptagon. Let me assume that the heptagon is regular with side length s, and the square also has side length s.

But since the square is constructed on the side AD, which is also a side of the heptagon, then the square is attached to the heptagon along AD. The Asymptote code draws A--C--D--F. So from A to C to D to F. If AD is a side of the square, then maybe AC and CD are sides of the square? Wait, but if AD is a side of the square, then the square would have sides AD, DC, CB, and BA. But in the Asymptote code, they draw A--C--D--F. So maybe C is adjacent to A and D? If AD is a side of the square, then if C is adjacent to A, then AC would be another side of the square. Similarly, CD would be another side. But then D is connected to F? Maybe F is another vertex of the square? But the Asymptote code also mentions point B, which is dir(360*2/7 + 90). Let me check that angle. 360*(2/7) is about 102.857 degrees, plus 90 is 192.857 degrees. So point B is at 192.857 degrees on the unit circle.

So maybe the square has vertices A, B, C, D? But then the Asymptote code draws A--C--D--F. Maybe F is another point? Wait, the Asymptote code draws the heptagon first, then defines points A, F, C, D, B. Then draws A--C--D--F. So maybe the square is A-C-D-F? But then A-C-D-F would be a square? Let me check. If AD is a side of the square, then AC and DF would be other sides? Wait, but if AD is a side, then the square would have sides AD, DC, CF, and FA? But then the Asymptote code draws A--C--D--F. So A to C, C to D, D to F, and then presumably F back to A? If that's the case, then A-C-D-F-A would be the square. So that would make sense. Then, since AD is a side of the square, then AC and DF would be sides of the square? Wait, but if AD is a side, then AC would be adjacent to AD? Wait, maybe not. Let me think.

Alternatively, maybe the square is A-B-C-D? But then the Asymptote code draws A--C--D--F. Maybe the square is A-C-D-F? If that's the case, then AC, CD, DF, and FA are sides of the square. But then FA would be a side. However, point F is a vertex of the heptagon. So FA would be a side of the square and also a side of the heptagon? Wait, but in the heptagon, adjacent vertices are connected by sides. If F is adjacent to A in the heptagon, then FA would be a side of the heptagon. But if FA is also a side of the square, then the square and heptagon share two sides? But the problem says they share a common side AD. So maybe FA is not a side of the square.

This is getting too confusing. Maybe I need to approach this with coordinates. Let me try to assign coordinates to the points.

Assume the heptagon is centered at the origin, with vertices on the unit circle. Let me compute coordinates for points A, D, F, B.

First, point A is at angle 90 + 360*(3/7) degrees. Let me compute that angle:

3/7 of 360 is 360*(3/7) ≈ 154.2857 degrees. Adding 90 degrees gives 244.2857 degrees. So point A is at 244.2857 degrees. Similarly, point D is... Wait, the problem says they share side AD. So if AD is a side of the heptagon, then D is the next vertex after A in the heptagon. Since the heptagon is drawn with vertices at angles starting from 90 degrees, increasing by 360/7 degrees each time. So if point A is at angle θ, then the next vertex (D) would be at θ + 360/7 degrees. Wait, but in the Asymptote code, point A is defined as dir(360*3/7 + 90). Let me check the indices. The Asymptote code draws the heptagon with vertices from i=0 to 7, so 8 points? Wait, no, for(int i=0; i <=7; ++i) would be 8 points, but a heptagon has 7 sides. Wait, maybe it's a typo? Or maybe it's drawing a heptagon with 7 sides, so i from 0 to 6? Maybe the Asymptote code has a mistake? But regardless, the code defines point A as dir(360*3/7 + 90). Let me think of the heptagon as having vertices labeled 0 through 6, each separated by 360/7 degrees. So starting from angle 90 degrees (which would be vertex 0), then vertex 1 is at 90 + 360/7 degrees, vertex 2 at 90 + 2*(360/7), etc. So point A is vertex 3: 90 + 3*(360/7) degrees. Then, the next vertex (vertex 4) would be at 90 + 4*(360/7) degrees, which is point F. Therefore, AD is the side from vertex 3 to vertex 4, which is the side between A and F? Wait, but in the Asymptote code, point D is defined as C + F - A. Wait, maybe there's confusion here.

Alternatively, maybe the Asymptote code defines point D as the next vertex after A in the heptagon? If A is vertex 3, then the next vertex is vertex 4 (point F). But then AD would be AF? But the problem says they share side AD. So maybe D is vertex 3 + 1? Wait, maybe I need to check the labels. The Asymptote code labels A, B, C, D, F. So maybe D is a different point?

Alternatively, maybe the square is constructed externally on the side AD of the heptagon. So AD is a side of both the heptagon and the square. Then, the square extends from AD in some direction. Since the heptagon is regular, the side AD is part of the heptagon, and the square is built on AD.

To find angle BAC, we need to figure out the positions of points A, B, and C.

Let me try to think of the square. If AD is a side of the square, then the square has sides AD, DC, CB, and BA. So from point A, moving along AD to D, then from D moving perpendicularly to AD to get to C, then from C moving back to B, and from B back to A. But since the square is in the plane with the heptagon, we need to figure out the direction of the square relative to the heptagon.

Alternatively, since the Asymptote code defines point C as A + dir(90)*(F - A). Let me think of this as moving from point A in the direction of 90 degrees (upwards) by the vector from A to F. Wait, if F is another point on the heptagon, then vector AF is from A to F. If we take that vector and scale it in the direction of 90 degrees, then add it to A, we get point C.

Alternatively, maybe dir(90) is a unit vector, so dir(90)*(F - A) is scaling the vector (F - A) by the direction of 90 degrees? Wait, no, in Asymptote, if you have a vector v, then dir(theta) * v would be a vector in the direction theta with magnitude equal to the magnitude of v? Wait, no, actually, dir(theta) is a unit vector, so multiplying by a vector would be a bit ambiguous. Wait, maybe the code is using vector operations. If you have a point A and a vector, then adding the vector to A gives another point. So dir(90) is a unit vector pointing upwards. So dir(90)*(F - A) would be scaling the vector (F - A) by the direction of 90 degrees? Wait, no, that doesn't make sense. Maybe it's a typo or misunderstanding.

Alternatively, maybe the code is trying to construct point C by moving from A in the direction perpendicular to AD? Since AD is a side of the square, then the square would have sides perpendicular to AD. If AD is a side of the heptagon, then the direction of AD can be determined.

Let me try to compute coordinates. Let me assume the heptagon is centered at the origin, with radius 1. Let me compute coordinates for points A and D (assuming D is the next vertex after A).

Point A is at angle θ = 90 + 3*(360/7) degrees. Let me compute θ:

First, 360/7 ≈ 51.4286 degrees. So 3*(360/7) ≈ 154.2857 degrees. Adding 90 degrees gives θ ≈ 244.2857 degrees. So coordinates of A are (cos(244.2857°), sin(244.2857°)). Similarly, the next vertex D would be at θ + 360/7 ≈ 244.2857 + 51.4286 ≈ 295.7143 degrees. So coordinates of D are (cos(295.7143°), sin(295.7143°)).

Now, vector AD is D - A. Let me compute that vector. But since both points are on the unit circle, the vector AD can be calculated. However, since we need to construct the square on AD, we need to find the direction perpendicular to AD.

In the square, the side AD is one side, and the adjacent sides are perpendicular to AD. So if we can find the direction of the square relative to the heptagon, we can find points B and C.

Alternatively, since the Asymptote code defines point C as A + dir(90)*(F - A), and point F is the next vertex after A in the heptagon. Wait, earlier I thought point F was the next vertex after A, which would be D? Wait, no, if point A is vertex 3, then the next vertex is vertex 4, which is F. Then D is another point? Wait, the Asymptote code defines D as C + F - A. So if I can figure out coordinates for C, then I can find D.

Alternatively, maybe I need to use complex numbers for easier calculations. Let me represent points as complex numbers. Let me denote the center of the heptagon as the origin.

Let me denote point A as a complex number: A = e^{iθ}, where θ = 90° + 3*(360°/7). Similarly, point F is the next vertex, so F = e^{i(θ + 360°/7)}.

Then, vector AF is F - A. The Asymptote code defines point C as A + dir(90)*(F - A). dir(90) is e^{i90°} = i. So multiplying by i rotates the vector AF by 90 degrees counterclockwise. Therefore, point C = A + i*(F - A).

Let me compute that.

C = A + i*(F - A) = A - i*A + i*F = A*(1 - i) + i*F.

Then, point D is defined as C + F - A. Let me compute that:

D = C + F - A = [A*(1 - i) + i*F] + F - A = A*(1 - i) - A + i*F + F = A*(1 - i - 1) + F*(i + 1) = A*(-i) + F*(1 + i).

So D = -i*A + (1 + i)*F.

But since A and F are points on the unit circle, maybe we can compute coordinates. However, this might get complicated. Alternatively, maybe there's a geometric interpretation.

Since AD is a side of the square, then the square has sides AD, DC, CB, and BA. If we can find coordinates of C and B, then we can compute angle BAC.

Alternatively, since we have point C defined as A + i*(F - A), which is a rotation of vector AF by 90 degrees counterclockwise and adding it to A. So if we imagine vector AF, rotating it 90 degrees counterclockwise gives a vector perpendicular to AF. Adding that to A gives point C.

But since AD is a side of the square, and AF is a side of the heptagon, which is adjacent to AD? Wait, if A is connected to D and to F in the heptagon, then AF is a diagonal? Wait, no, in a regular heptagon, each vertex is connected to its adjacent vertices. So if A is connected to D, then D is the next vertex after A, so F would be the vertex after D? Wait, no, if A is vertex 3, then D is vertex 4 (F?), and then the next vertex is vertex 5? Wait, this is confusing.

Wait, in the Asymptote code, point F is defined as dir(360*4/7 + 90). So if the first vertex is at i=0: 90 degrees, then i=1: 90 + 360/7, i=2: 90 + 2*360/7, i=3: 90 + 3*360/7 (which is point A), i=4: 90 + 4*360/7 (which is point F). Therefore, in the heptagon, the vertices are labeled 0 to 6, with A being vertex 3, F being vertex 4, and so on. Therefore, the side AD would be from vertex 3 to vertex 4 (A to F)? But the problem says they share side AD. So maybe D is vertex 4? But in the Asymptote code, point D is defined as C + F - A. So maybe D is not a vertex of the heptagon? Wait, but the problem says they share a common side AD, so AD must be a side of both the square and the heptagon. Therefore, AD is a side of the heptagon, meaning that D is the next vertex after A in the heptagon. Therefore, if A is vertex 3, then D is vertex 4 (which is point F). But then the Asymptote code defines point F as dir(360*4/7 + 90), which would be vertex 4. Therefore, maybe there's a typo in the Asymptote code, and point D is actually point F? Or maybe the label D is different?

Alternatively, maybe the Asymptote code has a mistake, and point D is supposed to be another point. Given that the problem says they share side AD, and in the Asymptote code, point D is defined as C + F - A, which is a different point.

This is getting too tangled. Maybe I need to approach this with angles.

Let me recall that in a regular heptagon, the central angle between two adjacent vertices is 360/7 degrees. The internal angle at each vertex is ((7-2)/7)*180 = 128.571 degrees.

Now, angle BAC is the angle at point A between points B and C. To find this angle, I need to know the positions of points B, A, and C relative to each other.

Assuming that AD is a side of both the square and the heptagon, then in the heptagon, the side AD is between vertices A and D. In the square, AD is a side, so the square extends from A to D, and then from D, the square goes in a direction perpendicular to AD.

Let me consider the square. If AD is a side of the square, then the square has sides AD, DC, CB, and BA. So from point A, moving to D, then turning 90 degrees to get to C, then to B, then back to A.

To find angle BAC, which is the angle between BA and CA.

But to find this angle, I need to know the positions of B and C relative to A.

Alternatively, since the square is constructed on AD, and the heptagon is regular, maybe we can find the angles between the sides.

Let me consider the heptagon first. Let me denote the center of the heptagon as O. The central angle between A and D is 360/7 degrees, since they are adjacent vertices.

Now, in the square, AD is a side, and the square is constructed such that from A, moving to D, then turning 90 degrees to get to C. The direction of this turn depends on whether the square is inside or outside the heptagon.

Assuming the square is constructed externally on the side AD, then the direction from D to C would be perpendicular to AD.

To find the angle BAC, we need to relate the positions of B and C with respect to A.

Alternatively, maybe using complex numbers would help. Let me assign complex numbers to points. Let me assume the heptagon is centered at the origin, with radius 1. Let me denote point A as a complex number, and point D as another complex number.

Let me denote the angle for point A as θ_A = 90° + 3*(360°/7). Similarly, point D is θ_D = θ_A + 360°/7.

The vector AD is D - A. In the square, the next side from D is DC, which is perpendicular to AD. So the direction from D to C is perpendicular to AD.

If we consider the square, then from D, moving in a direction perpendicular to AD to reach C. Similarly, from A, moving in a direction perpendicular to AD to reach B? Wait, no. In the square, from A, moving along AD to D, then from D, moving perpendicularly to AD to get to C, then from C, moving back towards B, and from B back to A.

Alternatively, since AD is a side of the square, then AB and AD are adjacent sides of the square, meaning that AB is perpendicular to AD. Wait, but if AD is a side of the square, then AB would be another side, so AB is perpendicular to AD. But in the problem, point B is a vertex of the heptagon. Wait, the Asymptote code defines point B as dir(360*2/7 + 90). Let me compute that angle: 360*(2/7) ≈ 102.857°, plus 90° is 192.857°, which is in the third quadrant.

So point B is located at 192.857°, which is between 180° and 270°, so southwest direction.

If I can figure out the relationship between points A, B, and C, maybe I can compute the angle.

Alternatively, since angle BAC is formed by points B, A, and C, and we need to find its measure.

Let me think about the positions. If AD is a side of the square, then AB is another side of the square? Or is AB a diagonal?

Wait, if AD is a side of the square, then AB could be another side if the square is constructed such that A is connected to B and D. But since the square has four sides, if AD is one side, then AB would be another side, making angle BAD 90 degrees.

But point B is a vertex of the heptagon. So if AB is a side of the square and also a side of the heptagon, then AB would be a side of both. But the problem says they share only side AD. Therefore, AB is not a side of the heptagon. Therefore, AB is a side of the square but not of the heptagon.

Therefore, the square has sides AD, DC, CB, and BA, where BA is not a side of the heptagon.

Therefore, to find angle BAC, which is the angle at A between BA and CA.

To find this angle, I need to find the angle between vectors AB and AC.

Since AB is a side of the square, and AC is a diagonal of the square? Wait, no. If AD is a side of the square, and AB is another side, then AC would be the diagonal from A to C, which would be the diagonal of the square. But in that case, angle BAC would be 45 degrees, but that seems too simple, and the answer is supposed to be a common fraction, which might not be 45 degrees.

Wait, but maybe not. Let me think again. If AD is a side of the square, and AB is another side, then AB is perpendicular to AD. So angle BAD is 90 degrees. Then, if AC is the diagonal from A to C, then angle BAC would be half of 90 degrees, which is 45 degrees. But this seems too straightforward, and the problem involves a heptagon, so maybe there's more to it.

Alternatively, maybe AC is not the diagonal of the square. If the square is A-D-C-B-A, then AC would be a diagonal, yes. But if AB is a side, then angle BAC would be 45 degrees. But since the heptagon is involved, maybe the position of point C is different?

Wait, but according to the Asymptote code, point C is defined as A + dir(90)*(F - A). If F is the next vertex after A in the heptagon, then vector AF is a side of the heptagon. Rotating that vector by 90 degrees and adding to A gives point C.

If I consider that, then AC is a vector obtained by rotating AF by 90 degrees. So the angle between AF and AC is 90 degrees.

But AF is a side of the heptagon, and AC is a side of the square? Or is it?

Alternatively, since AD is a side of the square, and AF is a side of the heptagon, which is adjacent to AD? Wait, if A is connected to D and to F in the heptagon, then AF is a side of the heptagon, and AD is also a side. But in a regular heptagon, each vertex is connected to two adjacent vertices. So if AD is a side, then A is connected to D and to another vertex, say, let's call it E. Wait, but in the Asymptote code, point B is defined as dir(360*2/7 + 90). Let me check the positions again.

If the heptagon has vertices labeled 0 to 6, with vertex 0 at 90°, then vertex 1 is at 90° + 360/7°, vertex 2 at 90° + 2*360/7°, vertex 3 at 90° + 3*360/7° (which is point A), vertex 4 at 90° + 4*360/7° (point F), vertex 5 at 90° + 5*360/7°, vertex 6 at 90° + 6*360/7°, and back to vertex 0.

So, vertex A is vertex 3, connected to vertex 2 and vertex 4. Therefore, in the heptagon, point A is connected to vertex 2 (which would be point B?) and vertex 4 (which is point F). Wait, the Asymptote code defines point B as dir(360*2/7 + 90), which is vertex 2. So yes, point B is vertex 2, connected to A (vertex 3) and vertex 1.

Therefore, in the heptagon, side AB is between vertex 2 and 3, which is a side of the heptagon. But the problem says that the square and heptagon share side AD. So AD is a side of both, meaning that AD is a side of the heptagon (between vertex 3 and 4) and also a side of the square.

Therefore, the square has side AD, and since AB is a side of the heptagon, it is not a side of the square. Therefore, the square is constructed on side AD, extending outward from the heptagon.

Now, to find angle BAC, which is the angle at A between points B (vertex 2 of the heptagon) and C (a point defined in the Asymptote code).

To find this angle, I need to determine the positions of points B and C relative to A.

Since point B is vertex 2 of the heptagon, and point A is vertex 3, the angle between AB and the center is known. Similarly, point C is defined as A + dir(90)*(F - A). Let me try to compute this.

Let me consider complex numbers again. Let me denote the center of the heptagon as the origin, and all points on the unit circle.

Let me denote:

- Point A: e^{iθ}, where θ = 90° + 3*(360°/7)
- Point F: e^{i(θ + 360°/7)} (next vertex)
- Point B: e^{i(θ - 360°/7)} (previous vertex, since A is vertex 3, B is vertex 2)

Wait, if θ is the angle for vertex 3, then vertex 2 is θ - 360°/7, and vertex 4 is θ + 360°/7.

So, vector AF is F - A = e^{i(θ + 360°/7)} - e^{iθ} = e^{iθ}(e^{i360°/7} - 1).

Then, dir(90°) is e^{i90°} = i. So point C = A + i*(F - A) = e^{iθ} + i*(e^{iθ}(e^{i360°/7} - 1)) = e^{iθ}[1 + i(e^{i360°/7} - 1)].

This seems complex, but maybe we can simplify. Let me compute the term inside the brackets:

1 + i(e^{iα} - 1), where α = 360°/7.

Let me write e^{iα} = cosα + i sinα. Then:

i(e^{iα} - 1) = i(cosα + i sinα - 1) = i(cosα - 1) + i^2 sinα = i(cosα - 1) - sinα.

Therefore, the entire expression becomes:

1 - sinα + i(cosα - 1).

So point C = e^{iθ} * [1 - sinα + i(cosα - 1)].

This is getting complicated, but maybe we can find the angle of point C relative to A.

Alternatively, since we need angle BAC, which is the angle between vectors AB and AC.

Vector AB is B - A = e^{i(θ - α)} - e^{iθ} = e^{iθ}(e^{-iα} - 1).

Vector AC is C - A = [A + i(F - A)] - A = i(F - A) = i(e^{iθ}(e^{iα} - 1)) = e^{iθ} * i(e^{iα} - 1).

So vector AC = e^{iθ} * i(e^{iα} - 1).

Therefore, the angle between AB and AC can be found by taking the argument of (AC / AB).

Let me compute AC / AB:

AC / AB = [e^{iθ} * i(e^{iα} - 1)] / [e^{iθ}(e^{-iα} - 1)] = [i(e^{iα} - 1)] / [e^{-iα} - 1].

Simplify denominator: e^{-iα} - 1 = - (1 - e^{-iα}) = -e^{-iα/2} * 2i sin(α/2) [using identity 1 - e^{-iα} = 2i sin(α/2) e^{-iα/2}]. Wait, let me recall that 1 - e^{-iα} = e^{-iα/2} * 2i sin(α/2).

Wait, more carefully:

Let me write 1 - e^{-iα} = e^{-iα/2} * [e^{iα/2} - e^{-iα/2}] = e^{-iα/2} * 2i sin(α/2).

Therefore, e^{-iα} - 1 = - (1 - e^{-iα}) = -e^{-iα/2} * 2i sin(α/2).

Similarly, numerator: i(e^{iα} - 1) = i * [e^{iα} - 1] = i * e^{iα/2} * [e^{iα/2} - e^{-iα/2}] = i * e^{iα/2} * 2i sin(α/2) = -2 sin(α/2) * e^{iα/2}.

Therefore, AC / AB = [ -2 sin(α/2) * e^{iα/2} ] / [ -e^{-iα/2} * 2i sin(α/2) ] = [ -2 sin(α/2) e^{iα/2} ] / [ -2i sin(α/2) e^{-iα/2} ] = [ e^{iα/2} ] / [ i e^{-iα/2} ] = e^{iα} / i = -i e^{iα}.

Because dividing by i is multiplying by -i.

So AC / AB = -i e^{iα}.

Therefore, the argument of AC / AB is the argument of -i e^{iα}.

Let me compute this.

First, e^{iα} has argument α.

-i has argument -π/2 or 3π/2.

Therefore, the total argument is α - π/2.

Therefore, the angle between vectors AB and AC is α - π/2.

But α is 360°/7, which is approximately 51.4286°.

So α - π/2 is 51.4286° - 90° = -38.5714°, which is equivalent to 360° - 38.5714° = 321.4286°, but since angles are modulo 360°, the actual angle between AB and AC is the absolute value, which is 38.5714°, or 360/7 - 90°, but let me compute it properly.

Wait, the angle between AB and AC is the argument of AC / AB, which is α - π/2. But since angles are measured from AB to AC, we need to consider the direction.

But since we're dealing with vectors, the angle between AB and AC is the absolute value of the argument, or depending on orientation.

But let me convert α to degrees: α = 360°/7 ≈ 51.4286°.

So α - π/2 is 51.4286° - 90° = -38.5714°, which is equivalent to 360° - 38.5714° = 321.4286°, but the angle between two vectors is the smallest angle between them, so it's 38.5714°, which is 360/7 - 90° + 360°? Wait, no.

Wait, the angle between AB and AC is |α - π/2|, but since α is in degrees, let me convert π/2 to degrees: π/2 = 90°. So the angle is |α - 90°| = |360/7 - 90| degrees.

Compute 360/7 ≈ 51.4286°, so 51.4286° - 90° = -38.5714°, absolute value is 38.5714°, which is 38.5714°.

But 38.5714° is equal to 270/7°, since 270 divided by 7 is approximately 38.5714. Let me check: 7*38 = 266, 7*38.5714 ≈ 270. Yes, 270/7 = 38.5714°.

Therefore, the angle between AB and AC is 270/7 degrees.

But wait, angle BAC is this angle? Wait, the angle between AB and AC is the angle at A between AB and AC, which is exactly angle BAC.

Therefore, the measure of angle BAC is 270/7 degrees.

But let me verify this.

Wait, earlier steps said that AC / AB has an argument of α - π/2, which is 360/7 - 90°, which is negative, but the angle between vectors is the absolute value. However, depending on the orientation, it could be 90° - α. Wait, let me think again.

If the argument of AC / AB is α - π/2, then the angle from AB to AC is α - π/2. But since α is less than π/2 (since 360/7 ≈ 51.4° < 90°), this would be a negative angle, meaning that AC is obtained by rotating AB clockwise by π/2 - α. Therefore, the angle between AB and AC is π/2 - α.

Wait, this is a crucial point. If the argument is negative, it means the angle is measured clockwise. The actual angle between the vectors is the smallest angle between them, which would be π/2 - α.

So π/2 - α is 90° - 360/7° = (630/7 - 360/7)° = 270/7° ≈ 38.57°, which matches the previous result.

Therefore, angle BAC is 270/7 degrees.

But let me check if this makes sense.

In the heptagon, the internal angle at A is 128.57°, which is 5*180/7 ≈ 128.57°. If angle BAC is 270/7 ≈ 38.57°, then that seems plausible.

But let me check with another approach.

Since AD is a side of the square, and AB is a side of the heptagon. The angle between AB and AD can be calculated.

In the heptagon, the internal angle at A is 128.57°, which is the angle between AB and AD. Wait, no. In a regular heptagon, each internal angle is the angle between two adjacent sides. So at vertex A, the two adjacent sides are AB and AD. Therefore, the internal angle at A is the angle between AB and AD, which is 128.57°.

But angle BAC is the angle between AB and AC. If AC is a side of the square, then we need to relate AC to AD.

Since AD is a side of the square, and AC is another side or diagonal?

Wait, earlier we found that angle between AB and AC is 270/7°, which is approximately 38.57°, and the internal angle between AB and AD is 128.57°, so angle between AC and AD would be 128.57° - 38.57° = 90°, which makes sense because AD is a side of the square, and AC is a side of the square? Wait, no. If angle between AB and AC is 38.57°, and angle between AB and AD is 128.57°, then angle between AC and AD is 128.57° - 38.57° = 90°, which would mean that AC is perpendicular to AD, which is consistent with AC being a side of the square.

Yes! Because in the square, AD is a side, and AC would be another side if the square is constructed such that from A, moving along AB (which is a side of the heptagon) and AC (which is a side of the square). Wait, no. If AD is a side of the square, then AC would be a diagonal or another side.

Wait, if angle between AC and AD is 90°, then AC is perpendicular to AD, which would mean that AC is a side of the square. Since AD is a side of the square, then AC would be another side, making angle CAD 90°, which would mean that the square has sides AD and AC. But then AB is another side?

Wait, this is getting confusing. Let me think again.

If angle between AB and AC is 270/7°, and angle between AB and AD is 128.57°, then angle between AC and AD is 128.57° - 270/7°. Let me compute 128.57° in fractions.

Since internal angle of heptagon is 5*180/7 = 900/7 ≈ 128.57°.

So angle between AC and AD is 900/7 - 270/7 = 630/7 = 90°, which is 90 degrees.

Therefore, AC is perpendicular to AD, which makes sense because AD is a side of the square, and AC is another side of the square, making a right angle at A.

Therefore, angle BAC is 270/7 degrees, which is the answer.

But let me check if 270/7 is reducible. 270 divided by 7 is 38 and 4/7, so as a fraction, it's 270/7.

But the problem says "Express your answer as a common fraction." So 270/7 is already a common fraction.

But let me check if this is correct.

Alternatively, maybe I made a mistake in the sign. Let me verify the calculation of the angle between AB and AC.

We had AC / AB = -i e^{iα}, whose argument is α - π/2. But since we're dealing with vectors, the angle between AB and AC is the absolute value of that argument, or π - |argument|?

Wait, the argument of AC / AB is the angle from AB to AC. If it's negative, it means clockwise. The actual angle between them is the smallest angle, which would be π/2 - α, as π/2 is 90°, and α is 360/7°.

So π/2 - α = 90° - 360/7° = (630/7 - 360/7)° = 270/7°, which is approximately 38.57°, which is less than 90°, so it's the correct angle between AB and AC.

Therefore, angle BAC is 270/7 degrees.

But let me check with another approach.

Let me consider the regular heptagon and the square.

In the regular heptagon, the central angles are 360/7°. The internal angle at each vertex is 5*180/7° = 900/7°.

At vertex A, the two adjacent vertices are B and D. The angle between AB and AD is 900/7°.

Now, the square has side AD, and from point A, the square extends in a direction perpendicular to AD. Since AD is a side of the square, the square's side AC is perpendicular to AD.

Therefore, angle between AD and AC is 90°, as it's a square.

Therefore, angle between AB and AC is angle between AB and AD minus angle between AC and AD, which is 900/7° - 90° = 900/7° - 630/7° = 270/7°, which matches our previous result.

Therefore, angle BAC is 270/7 degrees.

But let me check if this is the answer.

Wait, 270 divided by 7 is approximately 38.57°, which seems reasonable.

But let me check if there's an alternative way to get this.

Alternatively, since angle BAC is the difference between the internal angle of the heptagon at A and the right angle of the square.

Yes, as above.

Therefore, the measure of angle BAC is 270/7 degrees.

But let me check if the problem says "degree measure," so 270/7 is the answer.

But let me check if 270/7 reduces or if there's a miscalculation.

Wait, 270/7 is already in simplest terms.

But let me check with actual values.

Let me compute 270 divided by 7: 7*38 = 266, 270 - 266 = 4, so 270/7 = 38 4/7 degrees.

Alternatively, maybe the answer is expected in another form?

Wait, the problem says "degree measure," so 270/7 is acceptable as a common fraction.

But let me check if there's a different approach that might give a different answer.

Alternatively, using coordinates. Let me try to compute coordinates for points A, B, and C.

Assume the heptagon is unit circle.

Let me compute coordinates for point A, B, and C.

First, compute α = 360/7 ≈ 51.4286°.

Point A is at angle θ = 90° + 3α.

Let me compute θ: 90° + 3*(360/7)° = 90° + 1080/7° = 90° + 154.2857° ≈ 244.2857°.

Coordinates of A: (cos(244.2857°), sin(244.2857°)).

Similarly, point B is at angle θ - α = 244.2857° - 51.4286° ≈ 192.8571°.

Coordinates of B: (cos(192.8571°), sin(192.8571°)).

Point C is defined as A + dir(90)*(F - A).

Point F is at angle θ + α ≈ 244.2857° + 51.4286° ≈ 295.7143°.

Coordinates of F: (cos(295.7143°), sin(295.7143°)).

Vector AF is F - A.

dir(90) is (0,1). So dir(90)*(F - A) is (0,1) * (F_x - A_x, F_y - A_y) = (0, F_y - A_y).

Wait, no. If dir(90) is a unit vector, then multiplying by the vector (F - A) would be scaling? Wait, no, in Asymptote, if you have a vector v, then dir(theta) * v would be a vector in direction theta with magnitude equal to |v|. But I think in Asymptote, the * operator for vectors is component-wise multiplication, but I might be mistaken.

Alternatively, if dir(90) is a unit vector, then dir(90)*(F - A) would be the vector (F - A) rotated by 90 degrees counterclockwise.

Yes, that makes sense. Because multiplying by i in complex numbers corresponds to rotating by 90 degrees.

So if we consider F - A as a complex number, multiplying by i rotates it 90 degrees counterclockwise.

Therefore, point C = A + i*(F - A).

Let me compute this in complex numbers.

Let me denote A as a complex number: A = e^{iθ}, F = e^{i(θ + α)}.

Then, F - A = e^{i(θ + α)} - e^{iθ} = e^{iθ}(e^{iα} - 1).

Multiplying by i: i*(F - A) = i*e^{iθ}(e^{iα} - 1).

Adding to A: C = A + i*(F - A) = e^{iθ} + i*e^{iθ}(e^{iα} - 1) = e^{iθ}[1 + i(e^{iα} - 1)].

As before.

Now, to find angle BAC, we can compute the angle between vectors AB and AC.

Vector AB = B - A = e^{i(θ - α)} - e^{iθ} = e^{iθ}(e^{-iα} - 1).

Vector AC = C - A = i*(F - A) = i*e^{iθ}(e^{iα} - 1).

So, as before, the angle between AB and AC is the argument of (AC / AB) = [i(e^{iα} - 1)] / [e^{-iα} - 1].

As previously calculated, this simplifies to -i e^{iα}, whose argument is α - π/2, leading to angle BAC = π/2 - α = 90° - 360/7° = 270/7°.

Therefore, this confirms the previous result.

Thus, the measure of angle BAC is 270/7 degrees.

But let me check if this answer makes sense with the diagram.

In the Asymptote code, the square is drawn as A--C--D--F. So from A to C to D to F. If AD is a side of the square, then AC and CD are sides of the square. Therefore, angle at A between AC and AD is 90°, as it's a square.

Since angle between AB and AD is the internal angle of the heptagon, which is 900/7°, then angle between AB and AC is 900/7° - 90° = 270/7°, which matches.

Therefore, I'm confident that the answer is 270/7 degrees.

**Final Answer**
\boxed{\dfrac{270}{7}}

To determine the degree measure of angle $ \angle BAC $, we analyze the geometric configuration involving a square and a regular heptagon that share a common side $ \overline{AD} $.

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### Step 1: Understand the Geometry

- **Regular Heptagon**: A regular heptagon has 7 equal sides and 7 equal internal angles. The internal angle at each vertex is:
  $$
  \text{Internal angle} = \frac{(7 - 2) \cdot 180^\circ}{7} = \frac{900^\circ}{7}
  $$

- **Square**: A square has four equal sides and four right angles (90°). Since the square and the heptagon share side $ \overline{AD} $, this side is common to both.

- **Point A** is a vertex of both the square and the heptagon. The square extends from $ A $ to $ D $, and from $ A $, the square continues in a direction perpendicular to $ \overline{AD} $, forming a right angle at $ A $.

- **Point B** is a vertex of the heptagon adjacent to $ A $, and **Point C** is a vertex of the square adjacent to $ A $, forming the angle $ \angle BAC $.

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### Step 2: Use Geometric Relationships

- The internal angle at vertex $ A $ of the heptagon is $ \frac{900^\circ}{7} $, which is the angle between the sides $ \overline{AB} $ and $ \overline{AD} $.

- Since $ \overline{AD} $ is a side of the square, and the square has a right angle at $ A $, the angle between $ \overline{AD} $ and $ \overline{AC} $ is $ 90^\circ $.

- Therefore, the angle $ \angle BAC $ is the difference between the internal angle of the heptagon at $ A $ and the right angle of the square:
  $$
  \angle BAC = \frac{900^\circ}{7} - 90^\circ = \frac{900^\circ - 630^\circ}{7} = \frac{270^\circ}{7}
  $$

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### Final Answer

$$
\boxed{\dfrac{270}{7}}
$$

Solution: The measure of each interior angle in a regular $n$-gon is $180(n-2)/n$ degrees.  Therefore, the measure of angle $\angle BAD$ is $180(7-2)/7=\frac{900}7$ degrees and the measure of angle $CAD$ is 90 degrees.  Their difference, $\angle BAC$, measures  \[\frac{900}7-\frac{630}7=\boxed{\frac{270}7\text{ degrees}}.\]