Explanation:
Okay, so I need to find the measure of angle α in this hexagon inscribed in a circle. Let me start by recalling some properties of cyclic polygons. Since the hexagon is inscribed in a circle, it's a cyclic hexagon. I remember that in cyclic polygons, the opposite angles have some relationship, but I'm not sure exactly what. Wait, for cyclic quadrilaterals, opposite angles are supplementary. But this is a hexagon, so maybe there's a similar property?

Alternatively, maybe I can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc. That seems more useful here. Let me try to visualize the diagram. The Asymptote code defines points A through F around a circle centered at O. The labels mention angles at points F, B, and D. Specifically, angle at F is labeled 105 degrees, angle at B is 110 degrees, and angle α is at point D. The hexagon is drawn in the order A-B-C-D-E-F-A.

First, let me note down the given angles. At point F, the angle is 105°, and at point B, it's 110°. The angle we need to find is at point D, labeled α. Since it's a cyclic hexagon, all the vertices lie on the circle.

I need to relate the angles of the hexagon to the arcs between the points. For any polygon inscribed in a circle, the measure of an interior angle can be related to the arcs intercepted by the sides. Wait, for a cyclic polygon, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides? Or is it something else?

Let me think. In a cyclic quadrilateral, each interior angle is half the sum of the measures of the arcs intercepted by the two opposite angles. Wait, no. For a cyclic quadrilateral, the opposite angles are supplementary. Each angle is half the measure of the arc opposite to it. Wait, maybe I need to recall the formula for the interior angle of a cyclic polygon.

Alternatively, maybe I can think of each interior angle as half the difference of the intercepted arcs? Wait, no. Let me recall that an inscribed angle is half the measure of its intercepted arc. So, if I can figure out the arcs between the points, maybe I can find the angles.

But since it's a hexagon, there are six arcs between the six points. Let me denote the arcs between consecutive points as follows: arc AB, arc BC, arc CD, arc DE, arc EF, and arc FA. Let me denote their measures as a, b, c, d, e, f respectively. Since the total circumference is 360°, we have a + b + c + d + e + f = 360°.

Now, each interior angle of the hexagon can be related to these arcs. For example, the interior angle at point A is formed by the sides FA and AB. The measure of angle A would be half the sum of the arcs intercepted by the opposite arcs? Wait, no. Wait, in a cyclic polygon, the interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides? Wait, maybe not. Let me think again.

Actually, for any polygon, the interior angle can be calculated if we know the arcs. But for cyclic polygons, there's a relationship between the angles and the arcs. Let me consider triangle AOB, where O is the center. Wait, but maybe that's complicating things.

Alternatively, consider that each interior angle of the cyclic polygon is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. Let me think of a triangle. In a triangle inscribed in a circle, the angle is half the measure of the intercepted arc. For quadrilaterals, each angle is half the measure of the intercepted arc. Wait, no. For a cyclic quadrilateral, each angle is half the measure of the arc opposite to it. Wait, no. Let me recall:

In a cyclic quadrilateral, the opposite angles are supplementary. Each angle is half the measure of the arc opposite to it. Wait, if angle A is half the measure of arc BCD, and angle C is half the measure of arc BAD. Since arc BCD + arc BAD = 360°, then angle A + angle C = 180°, which is the property of cyclic quadrilaterals.

So, for a cyclic quadrilateral, each angle is half the measure of the arc opposite to it. For a cyclic polygon with more sides, maybe similar relationships hold?

But for a cyclic hexagon, maybe each interior angle is half the sum of the measures of the arcs intercepted by the two adjacent sides? Wait, no. Let me think again. Let me take a specific example. Let's take angle at point B. The angle at point B is formed by sides AB and BC. The intercepted arcs would be arc AC? Wait, no. Wait, in a triangle, the angle is half the intercepted arc. In a quadrilateral, the angle is half the intercepted arc opposite. Wait, maybe for a polygon with more sides, the interior angle is half the sum of the intercepted arcs?

Wait, perhaps I need to use the formula for the interior angle of a cyclic polygon. Let me check.

In general, for a cyclic polygon with n sides, the measure of an interior angle can be calculated if we know the arcs. However, since the polygon is cyclic, the sum of all interior angles is (n-2)*180°, which for a hexagon is (6-2)*180° = 720°. But how does that relate to the arcs?

Alternatively, maybe I can use the fact that each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, let me think. If I have an angle at vertex B, formed by chords AB and BC. The angle at B is an inscribed angle that intercepts arc AC. Wait, but in a triangle, the angle is half the intercepted arc. But in a polygon, the angle is formed by two adjacent sides, so maybe the angle is half the measure of the arc opposite to it?

Wait, maybe I need to think of the angle as formed by two chords, AB and BC. The angle at B is formed by those two chords. The measure of angle ABC is equal to half the measure of the arc that is opposite to it, which would be arc AC. Wait, but in a cyclic quadrilateral, angle ABC intercepts arc ADC, but in a hexagon, maybe it intercepts a different arc?

Wait, perhaps I need to recall that in any cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. Let me think again. For example, in a cyclic quadrilateral, angle ABC intercepts arc ADC. The measure of angle ABC is half the measure of arc ADC. Similarly, in a cyclic pentagon, angle ABC would intercept some arc?

Wait, maybe I need to use the formula for the interior angle of a cyclic polygon. Let me check.

Alternatively, since the hexagon is cyclic, all its vertices lie on the circle. Therefore, each interior angle can be related to the arcs between the vertices. Let me consider angle at point B. The angle at point B is formed by chords AB and BC. The measure of angle ABC is equal to half the measure of the arc that is opposite to angle B, which would be arc from A to C not containing B. Wait, but in a hexagon, there are two arcs between A and C: one going through B and the other going through the rest of the points. Since the hexagon is convex and inscribed in a circle, the angle at B would intercept the arc AC that does not contain B. Therefore, the measure of angle ABC is half the measure of arc AC.

Similarly, angle at point F is 105°, which would be half the measure of arc from E to A not containing F. Similarly, angle at point B is 110°, which is half the measure of arc from A to C not containing B. And angle α at point D is half the measure of arc from C to E not containing D.

Therefore, if I can find the measure of arc CE (not containing D), then α is half of that. Similarly, if I can find the measure of arc AC (not containing B), which is 2*110° = 220°, and arc EA (not containing F) is 2*105° = 210°. Wait, but how do I relate these arcs?

But since the total circumference is 360°, the arcs between the points add up to 360°. Let me denote the arcs between consecutive points as follows:

Arc AB = a

Arc BC = b

Arc CD = c

Arc DE = d

Arc EF = e

Arc FA = f

So, a + b + c + d + e + f = 360°

Now, angle at B is half the measure of arc AC (not containing B). Arc AC is arc AB + arc BC = a + b. Wait, but if angle at B intercepts arc AC not containing B, then arc AC would be the arc going from A to C through the other side of the circle, which would be arc AF + arc FE + arc ED + arc DC? Wait, no. Wait, if the angle is at point B, then the intercepted arc is the arc opposite to angle B, which would be arc AC that does not pass through B. Since the hexagon is labeled in order A-B-C-D-E-F-A, the arc AC that does not contain B would be the arc going from A to C through F, E, D? Wait, no. Wait, if you have points A, B, C, D, E, F in order around the circle, then arc AC that does not contain B would be the arc going from A to C passing through F, E, D? Wait, but that would be a long arc. Wait, but the measure of angle ABC is half the measure of arc AC that does not contain B. So, arc AC that does not contain B is the arc from A to C passing through F, E, D? Wait, but that would be arc AF + arc FE + arc ED + arc DC? Wait, no. Wait, arc from A to C not containing B would be arc A to C going the other way around the circle. Since the points are in order A, B, C, D, E, F, then going from A to C through B is the short arc, but the arc not containing B would be the long arc from A to C passing through F, E, D. Wait, but that would be arc AF + arc FE + arc ED + arc DC? Wait, no. From A to C through F, E, D would be arc AF + arc FE + arc ED + arc DC? Wait, no. From A to C through F, E, D would be arc AF + arc FE + arc ED + arc DC? Wait, but arc DC is from D to C, which is the same as arc CD in the opposite direction. Wait, maybe I need to think of the arc from A to C not containing B as arc A to F to E to D to C? Which would be arc AF + arc FE + arc ED + arc DC? Wait, but arc AF is f, arc FE is e, arc ED is d, arc DC is c. So total arc length would be f + e + d + c. But since the total circumference is a + b + c + d + e + f = 360°, then arc AC not containing B is (c + d + e + f). But angle at B is half of that. Wait, but angle at B is given as 110°, so:

Angle at B = 1/2 * (arc AC not containing B) = 1/2 * (c + d + e + f) = 110°

Similarly, angle at F is given as 105°, which is half the measure of arc EA not containing F. Let me check that. The angle at F is formed by sides EF and FA. The intercepted arc would be arc EA not containing F. Arc EA not containing F would be arc from E to A passing through B, C, D? Wait, points are A, B, C, D, E, F. So arc EA not containing F would be arc E to A through B, C, D. Which would be arc EB + arc BC + arc CD + arc DA? Wait, no. From E to A through B, C, D would be arc ED + arc DC + arc CB + arc BA? Wait, arc ED is d, arc DC is c, arc CB is b, arc BA is a. So total arc length is d + c + b + a. Therefore, angle at F is half of that. Therefore:

Angle at F = 1/2 * (a + b + c + d) = 105°

Similarly, angle at D is α, which is formed by sides CD and DE. The intercepted arc would be arc CE not containing D. Arc CE not containing D would be arc from C to E passing through F, A, B? Let me check. Points are A, B, C, D, E, F. So arc CE not containing D would be arc C to E through F, A, B. Which would be arc CF + arc FA + arc AB + arc BE? Wait, no. From C to E through F, A, B would be arc CF + arc FA + arc AB + arc BE? Wait, arc CF is arc from C to F, which is arc CD + arc DE + arc EF? Wait, no. Wait, arc from C to E through F, A, B would be arc C to F to A to B to E? Wait, that seems complicated. Wait, maybe it's easier to think that arc CE not containing D is the arc that goes from C to E without passing through D. Since the points are in order A, B, C, D, E, F, then arc CE not containing D would be the arc going from C to E through F, A, B? Wait, but that would pass through F, A, B. Wait, but from C to E, the arc that doesn't contain D would be the arc that goes the other way around the circle. Since D is between C and E, the arc CE that doesn't contain D would be the major arc CE passing through F, A, B. Therefore, the measure of that arc is arc CF + arc FA + arc AB + arc BE? Wait, no. Let me think again. From C to E, the arc not containing D would be the arc that goes from C to F to E? Wait, no. If you go from C to E through F, that would be arc C to F to E, which is arc CD + arc DE? Wait, no. Wait, arc from C to E through F would be arc C to D to E? Wait, no. Wait, if you go from C to E through F, you have to go from C to F to E. But arc CF is arc CD + arc DE + arc EF? Wait, no. Wait, arc CF is from C to F, which is arc CD + arc DE + arc EF? Wait, no. If the points are in order A, B, C, D, E, F, then arc from C to F would be arc CD + arc DE + arc EF. So that's c + d + e. Then arc from F to E is just arc FE, which is e? Wait, no. Wait, arc from C to E through F is arc CF + arc FE? Wait, arc CF is c + d + e? Wait, no. Wait, arc from C to F is arc CD + arc DE + arc EF? Wait, arc CD is c, arc DE is d, arc EF is e. So arc CF is c + d + e. Then arc FE is e? Wait, no. Arc FE is just e. Wait, but if you go from C to E through F, that would be arc CF + arc FE? But arc FE is e, and arc CF is c + d + e? Wait, that seems overlapping. Wait, maybe I need to think of the arc CE not containing D as the arc that goes from C to E passing through F, A, B. Which would be arc C to F to A to B to E. Let me calculate that. From C to F is arc CF, which is c + d + e. Then from F to A is arc FA, which is f. Then from A to B is arc AB, which is a. Then from B to E is arc BE, which is arc BC + arc CD + arc DE? Wait, arc BE would be arc BC + arc CD + arc DE? Wait, arc BC is b, arc CD is c, arc DE is d. So arc BE is b + c + d. Therefore, total arc CE not containing D would be arc CF + arc FA + arc AB + arc BE? Wait, that seems too long. Wait, but if you go from C to E through F, A, B, that would be arc C to F to A to B to E. Which is arc CF (c + d + e) + arc FA (f) + arc AB (a) + arc BE? Wait, but arc BE is from B to E, which is arc BC + arc CD + arc DE? Wait, but that would be b + c + d. However, if we already have arc CF (c + d + e) and arc FA (f) and arc AB (a), then adding arc BE would be overlapping? Wait, this seems confusing. Maybe there's a simpler way.

Alternatively, since the total circumference is 360°, the measure of arc CE not containing D is equal to 360° minus the measure of arc CE containing D. Arc CE containing D is arc CD + arc DE = c + d. Therefore, arc CE not containing D is 360° - (c + d). Therefore, angle at D, which is α, is half of that. Therefore:

α = 1/2 * (360° - (c + d)) = 180° - (c + d)/2

Similarly, earlier, we had for angle at B:

Angle at B = 1/2 * (arc AC not containing B) = 1/2 * (arc AF + arc FE + arc ED + arc DC) = 1/2 * (f + e + d + c) = 110°

Similarly, angle at F = 1/2 * (arc EA not containing F) = 1/2 * (arc EB + arc BC + arc CD + arc DA) = 1/2 * (arc EB + arc BC + arc CD + arc DA). Wait, arc EA not containing F would be arc from E to A passing through B, C, D. Which is arc EB + arc BC + arc CD + arc DA? Wait, arc EB is arc from E to B, which is arc ED + arc DC + arc CB? Wait, arc ED is d, arc DC is c, arc CB is b. So arc EB is d + c + b. Then arc BC is b, arc CD is c, arc DA is arc from D to A, which is arc DE + arc EF + arc FA? Wait, arc DA would be arc DE + arc EF + arc FA? Wait, no. Arc DA is from D to A, which is arc DC + arc CB + arc BA? Wait, arc DC is c, arc CB is b, arc BA is a. So arc DA is c + b + a. Therefore, arc EA not containing F is arc EB + arc BC + arc CD + arc DA? Wait, that seems too long. Wait, maybe I need to think again.

Alternatively, since arc EA not containing F is the arc from E to A that doesn't pass through F. Since the points are in order A, B, C, D, E, F, the arc from E to A not containing F would be the arc going from E to D to C to B to A. Which is arc ED + arc DC + arc CB + arc BA. Which is d + c + b + a. Therefore, angle at F is 1/2*(a + b + c + d) = 105°, as I had earlier.

Similarly, angle at B is 1/2*(arc AC not containing B) = 1/2*(arc AF + arc FE + arc ED + arc DC) = 1/2*(f + e + d + c) = 110°, which is the same as 1/2*(c + d + e + f) = 110°. Since a + b + c + d + e + f = 360°, then c + d + e + f = 360° - a - b. Therefore, angle at B is 1/2*(360° - a - b) = 110°, so 360° - a - b = 220°, so a + b = 360° - 220° = 140°. Therefore, a + b = 140°.

Similarly, angle at F is 1/2*(a + b + c + d) = 105°, so a + b + c + d = 210°. Since we already know that a + b = 140°, then c + d = 210° - 140° = 70°. Therefore, c + d = 70°.

Earlier, we had that α = 180° - (c + d)/2. Since c + d = 70°, then α = 180° - 70°/2 = 180° - 35° = 145°. Therefore, α is 145 degrees. Wait, is that correct? Let me check again.

Wait, let me recap:

From angle at B: 1/2*(c + d + e + f) = 110°, so c + d + e + f = 220°. Since total circumference is 360°, then a + b = 360° - 220° = 140°, which is correct.

From angle at F: 1/2*(a + b + c + d) = 105°, so a + b + c + d = 210°. Since a + b = 140°, then c + d = 210° - 140° = 70°, which is correct.

Then, for angle at D, which is α, we said that α = 1/2*(arc CE not containing D) = 1/2*(360° - arc CE containing D). Arc CE containing D is arc CD + arc DE = c + d. Therefore, arc CE not containing D is 360° - (c + d) = 360° - 70° = 290°, so α = 1/2 * 290° = 145°. Therefore, α is 145 degrees. That seems to check out.

But let me verify if my reasoning for angle at D is correct. The angle at D is formed by sides CD and DE. The intercepted arc is the arc opposite to angle D, which would be arc CE not containing D. Since angle D is at point D, the intercepted arc is the arc that is opposite, which is arc CE. Since the hexagon is convex and inscribed in a circle, the angle at D intercepts arc CE that does not contain D. Therefore, the measure of angle D is half the measure of arc CE. Since arc CE is 360° - (arc CD + arc DE) = 360° - (c + d) = 360° - 70° = 290°, so angle D is 290° / 2 = 145°, which is α. Therefore, the measure of α is 145 degrees.

But let me check if there are other angles or arcs that I need to consider. For example, maybe there are other angles given? The problem mentions labels at F, B, and D. The Asymptote code labels 105° at F, 110° at B, and α at D. So we have two angles given, and we need to find the third. Since we used those two angles to find c + d = 70°, and then used that to find α. Seems okay.

Alternatively, maybe I can check if the sum of all interior angles of the hexagon is 720°, as it should be. Let me see. The sum of interior angles of a hexagon is (6-2)*180° = 720°. If I can compute all angles, they should add up to 720°. However, we only have three angles given: 105°, 110°, and α = 145°. The other three angles are at points A, C, E. Let me see if I can find them.

But maybe that's overcomplicating. Alternatively, let me check if my calculation for α is consistent. If α is 145°, then let's see:

We have angle at B: 110°, angle at F: 105°, angle at D: 145°. Let me try to find the other angles.

For example, angle at A: formed by sides FA and AB. The intercepted arc would be arc opposite to A, which is arc from F to B not containing A. Arc FB not containing A would be arc FE + arc ED + arc DC + arc CB. Which is e + d + c + b. Since e + d + c + b = (b + c + d + e) = 210°, as found earlier. Therefore, angle at A is 1/2*(210°) = 105°. Wait, angle at A is 105°? But angle at F is also 105°. Interesting.

Similarly, angle at C: formed by sides BC and CD. The intercepted arc is arc from B to D not containing C. Arc BD not containing C would be arc BA + arc AF + arc FE + arc ED. Which is a + f + e + d. Since a + f + e + d = (a + b + c + d + e + f) - b - c = 360° - b - c. Wait, but we know that a + b = 140°, so a = 140° - b. Also, c + d = 70°, so c = 70° - d. Therefore, a + f + e + d = (140° - b) + f + e + d. But since a + b + c + d = 210°, and a + b = 140°, then c + d = 70°, so e + f = 360° - (a + b + c + d) = 360° - 210° = 150°. Therefore, e + f = 150°. Therefore, a + f + e + d = (140° - b) + (e + f) + d = 140° - b + 150° + d. But since c + d = 70°, and c = 70° - d, so maybe we need another relation. Wait, this seems getting too complicated. Maybe there's a better way.

Alternatively, since angle at C is half the measure of arc BD not containing C. Arc BD not containing C is arc from B to D passing through A, F, E. Which is arc BA + arc AF + arc FE + arc ED. Which is a + f + e + d. But since a + b + c + d = 210°, and a + b = 140°, then c + d = 70°, so e + f = 360° - (a + b + c + d) = 360° - 210° = 150°, as before. Therefore, arc BD not containing C is a + f + e + d = a + (e + f) + d = a + 150° + d. But since a + d = ?

Wait, we know that a + b = 140°, and c + d = 70°, but we don't know individual values of a, b, c, d, e, f. However, maybe we can find angle at C. Wait, angle at C is 1/2*(arc BD not containing C). But arc BD not containing C is equal to arc BA + arc AF + arc FE + arc ED = a + f + e + d. But since e + f = 150°, then arc BD not containing C is a + d + 150°. But we don't know a + d. However, since a + b = 140°, and c + d = 70°, but we don't have relations between a and d. Therefore, maybe this approach isn't helpful.

Alternatively, maybe I can find angle at E. Let me try. Angle at E is formed by sides DE and EF. The intercepted arc is arc from D to F not containing E. Arc DF not containing E would be arc DC + arc CB + arc BA + arc AF. Which is c + b + a + f. Which is (a + b + c) + f. Since a + b = 140°, and c = 70° - d, so a + b + c = 140° + 70° - d = 210° - d. Then adding f, which is part of e + f = 150°, so f = 150° - e. Therefore, arc DF not containing E is 210° - d + 150° - e = 360° - (d + e). But since c + d = 70°, and c = 70° - d, and e is part of e + f = 150°, but I don't know e. This seems too tangled.

Alternatively, maybe I can use the fact that the sum of all interior angles is 720°. If I can find the other angles, maybe I can check if they add up. Let me assume that angle at A is 105°, angle at B is 110°, angle at C is ?, angle at D is 145°, angle at E is ?, angle at F is 105°. So total so far is 105 + 110 + 145 + 105 = 465°. Therefore, the remaining angles at C and E must add up to 720 - 465 = 255°. If I can find angle at C and angle at E, maybe they add up to 255°. But without more information, it's hard to check. However, maybe there's another way.

Alternatively, maybe I can find angle at A. Earlier, I thought angle at A is 1/2*(arc FB not containing A). Arc FB not containing A is arc FE + arc ED + arc DC + arc CB = e + d + c + b. Which is e + d + c + b = (b + c + d + e) = 210°, as found earlier. Therefore, angle at A is 1/2*210° = 105°, which matches the angle at F. So angle at A is 105°, angle at F is 105°, angle at B is 110°, angle at D is 145°, and angles at C and E are unknown. If I can find another angle, maybe angle at C or E, I can check.

Let me try angle at E. Angle at E is formed by sides DE and EF. The intercepted arc is arc from D to F not containing E. Which is arc DC + arc CB + arc BA + arc AF = c + b + a + f. Which is (a + b + c) + f. Since a + b = 140°, and c = 70° - d, then a + b + c = 140° + 70° - d = 210° - d. Then adding f, which is part of e + f = 150°, so f = 150° - e. Therefore, arc DF not containing E is 210° - d + 150° - e = 360° - (d + e). Therefore, angle at E is 1/2*(360° - (d + e)) = 180° - (d + e)/2. But since c + d = 70°, and c = 70° - d, and e is part of e + f = 150°, but we don't know e or f. However, if I can find another relation.

Alternatively, maybe angle at C. Angle at C is formed by sides BC and CD. The intercepted arc is arc from B to D not containing C. Which is arc BA + arc AF + arc FE + arc ED = a + f + e + d. Which is a + d + e + f. Since e + f = 150°, then this is a + d + 150°. Therefore, angle at C is 1/2*(a + d + 150°). But since a + b = 140°, and c + d = 70°, but we don't know individual values. However, if I can express a + d in terms of known quantities. Let me think. From a + b = 140°, so a = 140° - b. From c + d = 70°, so d = 70° - c. Therefore, a + d = 140° - b + 70° - c = 210° - (b + c). But since b + c is part of the total. Wait, but we don't know b + c. However, since a + b + c + d = 210°, and a + b = 140°, then c + d = 70°, which we already know. Therefore, angle at C is 1/2*(a + d + 150°) = 1/2*( (a + d) + 150° ). But a + d = 210° - (b + c). Wait, this seems not helpful.

Alternatively, maybe I can use the fact that angle at C and angle at E are related. If I can find another equation. Wait, maybe angle at E is similar to angle at A? Since angle at A is 105°, angle at F is 105°, maybe angle at E is similar? Not sure. Alternatively, maybe there's symmetry? But the hexagon isn't regular, since the angles are different.

Alternatively, maybe I can use the fact that the sum of all angles is 720°, so if I assume angle at C and angle at E add up to 255°, and if I can find another relation between them. For example, angle at C is 1/2*(arc BD not containing C) = 1/2*(a + f + e + d) = 1/2*(a + d + e + f). Since e + f = 150°, then angle at C is 1/2*(a + d + 150°). Similarly, angle at E is 1/2*(arc DF not containing E) = 1/2*(c + b + a + f) = 1/2*(a + b + c + f). Since a + b = 140°, and c = 70° - d, and f = 150° - e. But this seems too tangled.

Alternatively, maybe I can use the fact that angle at C and angle at E are related through other arcs. But honestly, this might not be necessary. Since we already found α = 145°, and the problem only asks for α, maybe my initial calculation is correct. Let me check once again.

Given:

Angle at B: 110° = 1/2*(arc AC not containing B) = 1/2*(c + d + e + f) => c + d + e + f = 220°

Angle at F: 105° = 1/2*(arc EA not containing F) = 1/2*(a + b + c + d) => a + b + c + d = 210°

Subtracting the two equations:

(c + d + e + f) - (a + b + c + d) = 220° - 210° => e + f - a - b = 10°

But since a + b = 140°, then e + f = 140° + 10° = 150°, which matches our previous result that e + f = 150°.

Then, for angle at D: α = 1/2*(arc CE not containing D) = 1/2*(360° - (c + d)) = 1/2*(360° - 70°) = 1/2*(290°) = 145°, which is consistent.

Therefore, unless there is a miscalculation here, α should be 145°. Let me check once again the logic for angle at D.

Angle at D is formed by sides CD and DE. The intercepted arc is the arc opposite to angle D, which is arc CE not containing D. Since the hexagon is convex and inscribed in a circle, the angle at D intercepts arc CE that does not pass through D. The measure of that arc is 360° minus the measure of arc CE that does pass through D. Arc CE passing through D is arc CD + arc DE = c + d = 70°, so arc CE not containing D is 360° - 70° = 290°, so angle D is half of that, which is 145°. Yes, that seems correct.

Therefore, I think the measure of α is 145 degrees.

**Final Answer**
\boxed{145}

To determine the measure of angle $ \alpha $ in the given cyclic hexagon, we utilize the key geometric property of **cyclic polygons**: the measure of an **interior angle** is **half the measure of the arc** that it **intercepts** and is **opposite** to it.

---

### Step 1: Understand the Structure

We are given a **cyclic hexagon** inscribed in a circle. The vertices are labeled $ A, B, C, D, E, F $ in order around the circle. The angles at points $ B $ and $ F $ are given:

- $ \angle B = 110^\circ $
- $ \angle F = 105^\circ $

We are to find the measure of angle $ \alpha = \angle D $.

---

### Step 2: Use the Cyclic Polygon Angle-Arc Relationship

In a cyclic polygon, the **interior angle** at a vertex is **half the measure of the arc** that is **opposite** to that angle (i.e., the arc that does **not** contain the vertex).

So:

- $ \angle B = \frac{1}{2} \times \text{arc } AC $ (not containing $ B $)
- $ \angle F = \frac{1}{2} \times \text{arc } EA $ (not containing $ F $)
- $ \alpha = \angle D = \frac{1}{2} \times \text{arc } CE $ (not containing $ D $)

---

### Step 3: Express the Arcs in Terms of Variables

Let the arcs between consecutive vertices be:

- $ a = \text{arc } AB $
- $ b = \text{arc } BC $
- $ c = \text{arc } CD $
- $ d = \text{arc } DE $
- $ e = \text{arc } EF $
- $ f = \text{arc } FA $

Then, the total circumference of the circle is:

$$
a + b + c + d + e + f = 360^\circ
$$

---

### Step 4: Use Given Angles to Set Up Equations

From the angle at $ B $:

$$
\angle B = \frac{1}{2} \times (f + e + d + c) = 110^\circ
\Rightarrow f + e + d + c = 220^\circ \tag{1}
$$

From the angle at $ F $:

$$
\angle F = \frac{1}{2} \times (a + b + c + d) = 105^\circ
\Rightarrow a + b + c + d = 210^\circ \tag{2}
$$

Subtracting equation (2) from equation (1):

$$
(f + e + d + c) - (a + b + c + d) = 220^\circ - 210^\circ
\Rightarrow f + e - a - b = 10^\circ \tag{3}
$$

From equation (2), we can isolate $ c + d $:

$$
c + d = 210^\circ - (a + b) \tag{4}
$$

---

### Step 5: Use the Angle at $ D $ to Find $ \alpha $

The angle at $ D $, $ \alpha $, intercepts arc $ CE $ that does **not** contain $ D $. This arc is the **remaining** part of the circle after subtracting arc $ CD + DE = c + d $:

$$
\text{arc } CE = 360^\circ - (c + d)
$$

So:

$$
\alpha = \frac{1}{2} \times (360^\circ - (c + d)) = 180^\circ - \frac{c + d}{2}
$$

From equation (4), $ c + d = 210^\circ - (a + b) $. But from equation (3), we know:

$$
f + e = a + b + 10^\circ
$$

And from equation (2), $ a + b + c + d = 210^\circ $, so:

$$
c + d = 210^\circ - (a + b)
$$

Substitute into the expression for $ \alpha $:

$$
\alpha = 180^\circ - \frac{210^\circ - (a + b)}{2}
$$

But from equation (3), $ f + e = a + b + 10^\circ $, and from equation (1), $ f + e + c + d = 220^\circ $, so:

$$
a + b + 10^\circ + c + d = 220^\circ
\Rightarrow a + b + c + d = 210^\circ
$$

So, $ c + d = 210^\circ - (a + b) $, and we can now compute:

$$
\alpha = 180^\circ - \frac{210^\circ - (a + b)}{2}
$$

But we don't need to compute $ a + b $ explicitly. Instead, we can use the fact that:

$$
c + d = 70^\circ \quad \text{(from earlier)}
$$

So:

$$
\alpha = 180^\circ - \frac{70^\circ}{2} = 180^\circ - 35^\circ = \boxed{145^\circ}
$$

---

### Final Answer

$$
\boxed{145}
$$

[Assistant]
Explanation:
Okay, so I need to find the measure of angle α in this hexagon inscribed in a circle. Let me start by recalling some properties of cyclic polygons. Since the hexagon is inscribed in a circle, it's a cyclic hexagon. I remember that in cyclic polygons, the opposite angles have some relationship, but I'm not sure exactly what. Wait, for cyclic quadrilaterals, opposite angles are supplementary. But this is a hexagon, so maybe there's a similar property?

Alternatively, maybe I can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc. That seems more useful here. Let me try to visualize the diagram. The Asymptote code defines points A through F around a circle centered at O. The labels mention angles at points F, B, and D. Specifically, angle at F is labeled 105 degrees, angle at B is 110 degrees, and angle α is at point D. The hexagon is drawn in the order A-B-C-D-E-F-A.

First, let me note down the given angles. At point F, the angle is 105°, and at point B, it's 110°. The angle we need to find is at point D, labeled α. Since it's a cyclic hexagon, all the vertices lie on the circle.

I need to relate the angles of the hexagon to the arcs between the points. For any polygon inscribed in a circle, the measure of an interior angle can be related to the arcs intercepted by the sides. Wait, for a cyclic polygon, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides? Or is it something else?

Let me think. In a cyclic quadrilateral, each interior angle is half the sum of the measures of the arcs intercepted by the two opposite angles. Wait, no. For a cyclic quadrilateral, the opposite angles are supplementary. Each angle is half the measure of the arc opposite to it. Wait, maybe I need to recall the formula for the interior angle of a cyclic polygon.

Alternatively, maybe I can think of each interior angle as half the difference of the intercepted arcs? Wait, no. Let me recall that an inscribed angle is half the measure of its intercepted arc. So, if I can figure out the arcs between the points, maybe I can find the angles.

But since it's a hexagon, there are six arcs between the six points. Let me denote the arcs between consecutive points as follows: arc AB, arc BC, arc CD, arc DE, arc EF, and arc FA. Let me denote their measures as a, b, c, d, e, f respectively. Since the total circumference is 360°, we have a + b + c + d + e + f = 360°.

Now, each interior angle of the hexagon can be related to these arcs. For example, the interior angle at point A is formed by the sides FA and AB. The measure of angle A would be half the sum of the arcs intercepted by the opposite arcs? Wait, no. Wait, in a cyclic polygon, the interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides? Wait, maybe not. Let me think again.

Actually, for any polygon, the interior angle can be calculated if we know the arcs. But for cyclic polygons, there's a relationship between the angles and the arcs. Let me consider triangle AOB, where O is the center. Wait, but maybe that's complicating things.

Alternatively, consider that each interior angle of the cyclic polygon is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. Let me think of a triangle. In a triangle inscribed in a circle, the angle is half the measure of the intercepted arc. For quadrilaterals, each angle is half the measure of the intercepted arc. Wait, no. For a cyclic quadrilateral, each angle is half the measure of the arc opposite to it. Wait, no. Let me recall:

In a cyclic quadrilateral, the opposite angles are supplementary. Each angle is half the measure of the arc opposite to it. Wait, if angle A is half the measure of arc BCD, and angle C is half the measure of arc BAD. Since arc BCD + arc BAD = 360°, then angle A + angle C = 180°, which is the property of cyclic quadrilaterals.

So, for a cyclic quadrilateral, each angle is half the measure of the arc opposite to it. For a cyclic polygon with more sides, maybe similar relationships hold?

But for a cyclic hexagon, maybe each interior angle is half the sum of the measures of the arcs intercepted by the two adjacent sides? Wait, no. Let me think again. Let me take a specific example. Let's take angle at point B. The angle at point B is formed by sides AB and BC. The intercepted arcs would be arc AC? Wait, no. Wait, in a triangle, the angle is half the intercepted arc. In a quadrilateral, the angle is half the intercepted arc opposite. Wait, maybe for a polygon with more sides, the interior angle is half the sum of the intercepted arcs?

Wait, perhaps I need to use the formula for the interior angle of a cyclic polygon. Let me check.

In general, for a cyclic polygon with n sides, the measure of an interior angle can be calculated if we know the arcs. However, since the polygon is cyclic, the sum of all interior angles is (n-2)*180°, which for a hexagon is (6-2)*180° = 720°. But how does that relate to the arcs?

Alternatively, maybe I can use the fact that each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, let me think. If I have an angle at vertex B, formed by chords AB and BC. The angle at B is an inscribed angle that intercepts arc AC. Wait, but in a triangle, the angle is half the intercepted arc. But in a polygon, the angle is formed by two adjacent sides, so maybe the angle is half the measure of the arc opposite to it?

Wait, maybe I need to think of the angle as formed by two chords, AB and BC. The angle at B is formed by those two chords. The measure of angle ABC is equal to half the measure of the arc that is opposite to it, which would be arc AC. Wait, but in a cyclic quadrilateral, angle ABC intercepts arc ADC, but in a hexagon, maybe it intercepts a different arc?

Wait, perhaps I need to recall that in any cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. Let me think again. For example, in a cyclic quadrilateral, angle ABC intercepts arc ADC. The measure of angle ABC is half the measure of arc ADC. Similarly, in a cyclic pentagon, angle ABC would intercept some arc?

Wait, maybe I need to use the formula for the interior angle of a cyclic polygon. Let me check.

Alternatively, since the hexagon is cyclic, all its vertices lie on the circle. Therefore, each interior angle can be related to the arcs between the vertices. Let me consider angle at point B. The angle at point B is formed by chords AB and BC. The measure of angle ABC is equal to half the measure of the arc that is opposite to angle B, which would be arc from A to C not containing B. Wait, but in a hexagon, there are two arcs between A and C: one going through B and the other going through the rest of the points. Since the hexagon is convex and inscribed in a circle, the angle at B would intercept the arc AC that does not contain B. Therefore, the measure of angle ABC is half the measure of arc AC.

Similarly, angle at point F is 105°, which would be half the measure of arc from E to A not containing F. Similarly, angle at point B is 110°, which is half the measure of arc from A to C not containing B. And angle α at point D is half the measure of arc from C to E not containing D.

Therefore, if I can find the measure of arc CE (not containing D), then α is half of that. Similarly, if I can find the measure of arc AC (not containing B), which is 2*110° = 220°, and arc EA (not containing F) is 2*105° = 210°. Wait, but how do I relate these arcs?

But since the total circumference is 360°, the arcs between the points add up to 360°. Let me denote the arcs between consecutive points as follows:

Arc AB = a

Arc BC = b

Arc CD = c

Arc DE = d

Arc EF = e

Arc FA = f

So, a + b + c + d + e + f = 360°

Now, angle at B is half the measure of arc AC (not containing B). Arc AC is arc AB + arc BC = a + b. Wait, but if angle at B intercepts arc AC not containing B, then arc AC would be the arc going from A to C through the other side of the circle, which would be arc AF + arc FE + arc ED + arc DC? Wait, no. Wait, if the angle is at point B, then the intercepted arc is the arc opposite to angle B, which would be arc AC that does not pass through B. Since the hexagon is labeled in order A-B-C-D-E-F-A, the arc AC that does not contain B would be the arc going from A to C through F, E, D? Wait, no. Wait, if you have points A, B, C, D, E, F in order around the circle, then arc AC that does not contain B would be the arc going from A to C passing through F, E, D? Wait, but that would be a long arc. Wait, but the measure of angle ABC is half the measure of arc AC that does not contain B. So, arc AC that does not contain B is the arc from A to C passing through F, E, D? Wait, but that would be arc AF + arc FE + arc ED + arc DC? Wait, no. Wait, arc from A to C not containing B would be arc A to C going the other way around the circle. Since the points are in order A, B, C, D, E, F, then going from A to C through B is the short arc, but the arc not containing B would be the long arc from A to C passing through F, E, D. Wait, but that would be arc AF + arc FE + arc ED + arc DC? Wait, no. From A to C through F, E, D would be arc AF + arc FE + arc ED + arc DC? Wait, no. From A to C through F, E, D would be arc AF + arc FE + arc ED + arc DC? Wait, but arc DC is from D to C, which is the same as arc CD in the opposite direction. Wait, maybe I need to think of the arc from A to C not containing B as arc A to F to E to D to C? Which would be arc AF + arc FE + arc ED + arc DC? Wait, but arc AF is f, arc FE is e, arc ED is d, arc DC is c. So total arc length would be f + e + d + c. But since the total circumference is a + b + c + d + e + f = 360°, then arc AC not containing B is (c + d + e + f). But angle at B is half of that. Wait, but angle at B is given as 110°, so:

Angle at B = 1/2 * (arc AC not containing B) = 1/2 * (c + d + e + f) = 110°

Similarly, angle at F is given as 105°, which is half the measure of arc EA not containing F. Let me check that. The angle at F is formed by sides EF and FA. The intercepted arc would be arc EA not containing F. Arc EA not containing F would be arc from E to A passing through B, C, D? Wait, points are A, B, C, D, E, F. So arc EA not containing F would be arc E to A through B, C, D. Which would be arc EB + arc BC + arc CD + arc DA? Wait, no. From E to A through B, C, D would be arc ED + arc DC + arc CB + arc BA? Wait, arc ED is d, arc DC is c, arc CB is b, arc BA is a. So total arc length is d + c + b + a. Therefore, angle at F is half of that. Therefore:

Angle at F = 1/2 * (a + b + c + d) = 105°

Similarly, angle at D is α, which is formed by sides CD and DE. The intercepted arc would be arc CE not containing D. Arc CE not containing D would be arc from C to E passing through F, A, B? Let me check. Points are A, B, C, D, E, F. So arc CE not containing D would be arc C to E through F, A, B. Which would be arc CF + arc FA + arc AB + arc BE? Wait, no. From C to E through F, A, B would be arc CF + arc FA + arc AB + arc BE? Wait, arc CF is arc from C to F, which is arc CD + arc DE + arc EF? Wait, no. Wait, arc from C to E through F, A, B would be arc C to F to A to B to E? Wait, that seems complicated. Wait, maybe it's easier to think that arc CE not containing D is the arc that goes from C to E without passing through D. Since the points are in order A, B, C, D, E, F, then arc CE not containing D would be the arc going from C to E through F, A, B? Wait, but that would pass through F, A, B. Wait, but from C to E, the arc that doesn't contain D would be the arc that goes the other way around the circle. Since D is between C and E, the arc CE that doesn't contain D would be the major arc CE passing through F, A, B. Therefore, the measure of that arc is arc CF + arc FA + arc AB + arc BE? Wait, no. Let me think again. From C to E, the arc not containing D would be the arc that goes from C to F to E? Wait, no. If you go from C to E through F, that would be arc C to F to E, which is arc CD + arc DE? Wait, no. Wait, arc from C to E through F would be arc C to D to E? Wait, no. Wait, if you go from C to E through F, you have to go from C to F to E. But arc CF is arc CD + arc DE + arc EF? Wait, no. Wait, arc CF is from C to F, which is arc CD + arc DE + arc EF? Wait, no. If the points are in order A, B, C, D, E, F, then arc from C to F would be arc CD + arc DE + arc EF. So that's c + d + e. Then arc from F to E is just arc FE, which is e? Wait, no. Wait, arc from C to E through F is arc CF + arc FE? Wait, arc CF is c + d + e? Wait, no. Wait, arc from C to F is arc CD + arc DE + arc EF? Wait, arc CD is c, arc DE is d, arc EF is e. So arc CF is c + d + e. Then arc FE is e? Wait, no. Arc FE is just e. Wait, but if you go from C to E through F, that would be arc CF + arc FE? But arc FE is e, and arc CF is c + d + e? Wait, that seems overlapping. Wait, maybe I need to think of the arc CE not containing D as the arc that goes from C to E passing through F, A, B. Which would be arc C to F to A to B to E. Let me calculate that. From C to F is arc CF, which is c + d + e. Then from F to A is arc FA, which is f. Then from A to B is arc AB, which is a. Then from B to E is arc BE, which is arc BC + arc CD + arc DE? Wait, arc BE would be arc BC + arc CD + arc DE? Wait, arc BC is b, arc CD is c, arc DE is d. So arc BE is b + c + d. Therefore, total arc CE not containing D would be arc CF + arc FA + arc AB + arc BE? Wait, that seems too long. Wait, but if you go from C to E through F, A, B, that would be arc C to F to A to B to E. Which is arc CF (c + d + e) + arc FA (f) + arc AB (a) + arc BE? Wait, but arc BE is from B to E, which is arc BC + arc CD + arc DE? Wait, but that would be b + c + d. However, if we already have arc CF (c + d + e) and arc FA (f) and arc AB (a), then adding arc BE would be overlapping? Wait, this seems confusing. Maybe there's a simpler way.

Alternatively, since the total circumference is 360°, the measure of arc CE not containing D is equal to 360° minus the measure of arc CE containing D. Arc CE containing D is arc CD + arc DE = c + d. Therefore, arc CE not containing D is 360° - (c + d). Therefore, angle at D, which is α, is half of that. Therefore:

α = 1/2 * (360° - (c + d)) = 180° - (c + d)/2

Similarly, earlier, we had for angle at B:

Angle at B = 1/2 * (arc AC not containing B) = 1/2 * (arc AF + arc FE + arc ED + arc DC) = 1/2 * (f + e + d + c) = 110°

Similarly, angle at F = 1/2 * (arc EA not containing F) = 1/2 * (arc EB + arc BC + arc CD + arc DA) = 1/2 * (arc EB + arc BC + arc CD + arc DA). Wait, arc EA not containing F would be arc from E to A passing through B, C, D. Which is arc EB + arc BC + arc CD + arc DA? Wait, arc EB is arc from E to B, which is arc ED + arc DC + arc CB? Wait, arc ED is d, arc DC is c, arc CB is b. So arc EB is d + c + b. Then arc BC is b, arc CD is c, arc DA is arc from D to A, which is arc DE + arc EF + arc FA? Wait, arc DA would be arc DE + arc EF + arc FA? Wait, no. Arc DA is from D to A, which is arc DC + arc CB + arc BA? Wait, arc DC is c, arc CB is b, arc BA is a. So arc DA is c + b + a. Therefore, arc EA not containing F is arc EB + arc BC + arc CD + arc DA? Wait, that seems too long. Wait, maybe I need to think again.

Alternatively, since arc EA not containing F is the arc from E to A that doesn't pass through F. Since the points are in order A, B, C, D, E, F, the arc from E to A not containing F would be the arc going from E to D to C to B to A. Which is arc ED + arc DC + arc CB + arc BA. Which is d + c + b + a. Therefore, angle at F is 1/2*(a + b + c + d) = 105°, as I had earlier.

Similarly, angle at B is 1/2*(arc AC not containing B) = 1/2*(arc AF + arc FE + arc ED + arc DC) = 1/2*(f + e + d + c) = 110°, which is the same as 1/2*(c + d + e + f) = 110°. Since a + b + c + d + e + f = 360°, then c + d + e + f = 360° - a - b. Therefore, angle at B is 1/2*(360° - a - b) = 110°, so 360° - a - b = 220°, so a + b = 360° - 220° = 140°. Therefore, a + b = 140°.

Similarly, angle at F is 1/2*(a + b + c + d) = 105°, so a + b + c + d = 210°. Since we already know that a + b = 140°, then c + d = 210° - 140° = 70°. Therefore, c + d = 70°.

Earlier, we had that α = 180° - (c + d)/2. Since c + d = 70°, then α = 180° - 70°/2 = 180° - 35° = 145°. Therefore, α is 145 degrees. Wait, is that correct? Let me check again.

Wait, let me recap:

From angle at B: 1/2*(c + d + e + f) = 110°, so c + d + e + f = 220°. Since total circumference is 360°, then a + b = 360° - 220° = 140°, which is correct.

From angle at F: 1/2*(a + b + c + d) = 105°, so a + b + c + d = 210°. Since a + b = 140°, then c + d = 210° - 140° = 70°, which is correct.

Then, for angle at D, which is α, we said that α = 1/2*(arc CE not containing D) = 1/2*(360° - arc CE containing D). Arc CE containing D is arc CD + arc DE = c + d. Therefore, arc CE not containing D is 360° - (c + d) = 360° - 70° = 290°, so α = 1/2 * 290° = 145°. Therefore, α is 145 degrees. That seems to check out.

But let me verify if my reasoning for angle at D is correct. The angle at D is formed by sides CD and DE. The intercepted arc is the arc opposite to angle D, which would be arc CE not containing D. Since angle D is at point D, the intercepted arc is the arc that is opposite, which is arc CE. Since the hexagon is convex and inscribed in a circle, the angle at D intercepts arc CE that does not contain D. Therefore, the measure of angle D is half the measure of arc CE. Since arc CE is 360° - (arc CD + arc DE) = 360° - (c + d) = 360° - 70° = 290°, so angle D is 290° / 2 = 145°, which is α. Therefore, the measure of α is 145 degrees.

But let me check if there are other angles or arcs that I need to consider. For example, maybe there are other angles given? The problem mentions labels at F, B, and D. The Asymptote code labels 105° at F, 110° at B, and α at D. So we have two angles given, and we need to find the third. Since we used those two angles to find c + d = 70°, and then used that to find α. Seems okay.

Alternatively, maybe I can check if the sum of all interior angles of the hexagon is 720°, as it should be. Let me see. The sum of interior angles of a hexagon is (6-2)*180° = 720°. If I can compute all angles, they should add up to 720°. However, we only have three angles given: 105°, 110°, and α = 145°. The other three angles are at points A, C, E. Let me see if I can find them.

But maybe that's overcomplicating. Alternatively, let me check if my calculation for α is consistent. If α is 145°, then let's see:

We have angle at B: 110°, angle at F: 105°, angle at D: 145°. Let me try to find the other angles.

For example, angle at A: formed by sides FA and AB. The intercepted arc would be arc opposite to A, which is arc from F to B not containing A. Arc FB not containing A would be arc FE + arc ED + arc DC + arc CB. Which is e + d + c + b. Since e + d + c + b = (b + c + d + e) = 210°, as found earlier. Therefore, angle at A is 1/2*(210°) = 105°. Wait, angle at A is 105°? But angle at F is also 105°. Interesting.

Similarly, angle at C: formed by sides BC and CD. The intercepted arc is arc from B to D not containing C. Arc BD not containing C would be arc BA + arc AF + arc FE + arc ED. Which is a + f + e + d. Since a + f + e + d = (a + b + c + d + e + f) - b - c = 360° - b - c. Wait, but we know that a + b = 140°, so a = 140° - b. Also, c + d = 70°, so c = 70° - d. Therefore, a + f + e + d = (140° - b) + f + e + d. But since a + b + c + d = 210°, and a + b = 140°, then c + d = 70°, so e + f = 360° - (a + b + c + d) = 360° - 210° = 150°. Therefore, e + f = 150°. Therefore, a + f + e + d = (140° - b) + (e + f) + d = 140° - b + 150° + d. But since c + d = 70°, and c = 70° - d, so maybe we need another relation. Wait, this seems getting too complicated. Maybe there's a better way.

Alternatively, since angle at C is half the measure of arc BD not containing C. Arc BD not containing C is arc from B to D passing through A, F, E. Which is arc BA + arc AF + arc FE + arc ED. Which is a + f + e + d. But since a + b + c + d = 210°, and a + b = 140°, then c + d = 70°, so e + f = 360° - (a + b + c + d) = 360° - 210° = 150°, as before. Therefore, arc BD not containing C is a + f + e + d = a + (e + f) + d = a + 150° + d. But since a + d = ?

Wait, we know that a + b = 140°, and c + d = 70°, but we don't know individual values of a, b, c, d, e, f. However, maybe we can find angle at C. Wait, angle at C is 1/2*(arc BD not containing C). But arc BD not containing C is equal to arc BA + arc AF + arc FE + arc ED = a + f + e + d. But since e + f = 150°, then arc BD not containing C is a + d + 150°. But we don't know a + d. However, since a + b = 140°, and c + d = 70°, but we don't have relations between a and d. Therefore, maybe this approach isn't helpful.

Alternatively, maybe I can find angle at E. Let me try. Angle at E is formed by sides DE and EF. The intercepted arc is arc from D to F not containing E. Arc DF not containing E would be arc DC + arc CB + arc BA + arc AF. Which is c + b + a + f. Which is (a + b + c) + f. Since a + b = 140°, and c = 70° - d, so a + b + c = 140° + 70° - d = 210° - d. Then adding f, which is part of e + f = 150°, so f = 150° - e. Therefore, arc DF not containing E is 210° - d + 150° - e = 360° - (d + e). But since c + d = 70°, and c = 70° - d, and e is part of e + f = 150°, but I don't know e. This seems too tangled.

Alternatively, maybe I can use the fact that the sum of all interior angles is 720°. If I can find the other angles, maybe I can check if they add up. Let me assume that angle at A is 105°, angle at B is 110°, angle at C is ?, angle at D is 145°, angle at E is ?, angle at F is 105°. So total so far is 105 + 110 + 145 + 105 = 465°. Therefore, the remaining angles at C and E must add up to 720 - 465 = 255°. If I can find angle at C and angle at E, maybe they add up to 255°. But without more information, it's hard to check. However, maybe there's another way.

Alternatively, maybe I can find angle at A. Earlier, I thought angle at A is 1/2*(arc FB not containing A). Arc FB not containing A is arc FE + arc ED + arc DC + arc CB = e + d + c + b. Which is e + d + c + b = (b + c + d + e) = 210°, as found earlier. Therefore, angle at A is 1/2*210° = 105°, which matches the angle at F. So angle at A is 105°, angle at F is 105°, angle at B is 110°, angle at D is 145°, and angles at C and E are unknown. If I can find another angle, maybe angle at C or E, I can check.

Let me try angle at E. Angle at E is formed by sides DE and EF. The intercepted arc is arc from D to F not containing E. Which is arc DC + arc CB + arc BA + arc AF = c + b + a + f. Which is (a + b + c) + f. Since a + b = 140°, and c = 70° - d, then a + b + c = 140° + 70° - d = 210° - d. Then adding f, which is part of e + f = 150°, so f = 150° - e. Therefore, arc DF not containing E is 210° - d + 150° - e = 360° - (d + e). Therefore, angle at E is 1/2*(360° - (d + e)) = 180° - (d + e)/2. But since c + d = 70°, and c = 70° - d, and e is part of e + f = 150°, but we don't know e or f. However, if I can find another relation.

Alternatively, maybe angle at C. Angle at C is formed by sides BC and CD. The intercepted arc is arc from B to D not containing C. Which is arc BA + arc AF + arc FE + arc ED = a + f + e + d. Which is a + d + e + f. Since e + f = 150°, then this is a + d + 150°. Therefore, angle at C is 1/2*(a + d + 150°). But since a + b = 140°, and c + d = 70°, but we don't know individual values. However, if I can express a + d in terms of known quantities. Let me think. From a + b = 140°, so a = 140° - b. From c + d = 70°, so d = 70° - c. Therefore, a + d = 140° - b + 70° - c = 210° - (b + c). But since b + c is part of the total. Wait, but we don't know b + c. However, since a + b + c + d = 210°, and a + b = 140°, then c + d = 70°, which we already know. Therefore, angle at C is 1/2*(a + d + 150°) = 1/2*( (a + d) + 150° ). But a + d = 210° - (b + c). Wait, this seems not helpful.

Alternatively, maybe I can use the fact that angle at C and angle at E are related. If I can find another equation. Wait, maybe angle at E is similar to angle at A? Since angle at A is 105°, angle at F is 105°, maybe angle at E is similar? Not sure. Alternatively, maybe there's symmetry? But the hexagon isn't regular, since the angles are different.

Alternatively, maybe I can use the fact that the sum of all angles is 720°, so if I assume angle at C and angle at E add up to 255°, and if I can find another relation between them. For example, angle at C is 1/2*(arc BD not containing C) = 1/2*(a + f + e + d) = 1/2*(a + d + e + f). Since e + f = 150°, then angle at C is 1/2*(a + d + 150°). Similarly, angle at E is 1/2*(arc DF not containing E) = 1/2*(c + b + a + f) = 1/2*(a + b + c + f). Since a + b = 140°, and c = 70° - d, and f = 150° - e. But this seems too tangled.

Alternatively, maybe I can use the fact that angle at C and angle at E are related through other arcs. But honestly, this might not be necessary. Since we already found α = 145°, and the problem only asks for α, maybe my initial calculation is correct. Let me check once again.

Given:

Angle at B: 110° = 1/2*(arc AC not containing B) = 1/2*(c + d + e + f) => c + d + e + f = 220°

Angle at F: 105° = 1/2*(arc EA not containing F) = 1/2*(a + b + c + d) => a + b + c + d = 210°

Subtracting the two equations:

(c + d + e + f) - (a + b + c + d) = 220° - 210° => e + f - a - b = 10°

But since a + b = 140°, then e + f = 140° + 10° = 150°, which matches our previous result that e + f = 150°.

Then, for angle at D: α = 1/2*(arc CE not containing D) = 1/2*(360° - (c + d)) = 1/2*(360° - 70°) = 1/2*(290°) = 145°, which is consistent.

Therefore, unless there is a miscalculation here, α should be 145°. Let me check once again the logic for angle at D.

Angle at D is formed by sides CD and DE. The intercepted arc is the arc opposite to angle D, which is arc CE not containing D. Since the hexagon is convex and inscribed in a circle, the angle at D intercepts arc CE that does not pass through D. The measure of that arc is 360° minus the measure of arc CE that does pass through D. Arc CE passing through D is arc CD + arc DE = c + d = 70°, so arc CE not containing D is 360° - 70° = 290°, so angle D is half of that, which is 145°. Yes, that seems correct.

Therefore, I think the measure of α is 145 degrees.

**Final Answer**
\boxed{145}

To determine the measure of angle $ \alpha $ in the given cyclic hexagon, we utilize the key geometric property of **cyclic polygons**: the measure of an **interior angle** is **half the measure of the arc** that it **intercepts** and is **opposite** to it.

---

### Step 1: Understand the Structure

We are given a **cyclic hexagon** inscribed in a circle. The vertices are labeled $ A, B, C, D, E, F $ in order around the circle. The angles at points $ B $ and $ F $ are given:

- $ \angle B = 110^\circ $
- $ \angle F = 105^\circ $

We are to find the measure of angle $ \alpha = \angle D $.

---

### Step 2: Use the Cyclic Polygon Angle-Arc Relationship

In a cyclic polygon, the **interior angle** at a vertex is **half the measure of the arc** that is **opposite** to that angle (i.e., the arc that does **not** contain the vertex).

So:

- $ \angle B = \frac{1}{2} \times \text{arc } AC $ (not containing $ B $)
- $ \angle F = \frac{1}{2} \times \text{arc } EA $ (not containing $ F $)
- $ \alpha = \angle D = \frac{1}{2} \times \text{arc } CE $ (not containing $ D $)

---

### Step 3: Express the Arcs in Terms of Variables

Let the arcs between consecutive vertices be:

- $ a = \text{arc } AB $
- $ b = \text{arc } BC $
- $ c = \text{arc } CD $
- $ d = \text{arc } DE $
- $ e = \text{arc } EF $
- $ f = \text{arc } FA $

Then, the total circumference of the circle is:

$$
a + b + c + d + e + f = 360^\circ
$$

---

### Step 4: Use Given Angles to Set Up Equations

From the angle at $ B $:

$$
\angle B = \frac{1}{2} \times (f + e + d + c) = 110^\circ
\Rightarrow f + e + d + c = 220^\circ \tag{1}
$$

From the angle at $ F $:

$$
\angle F = \frac{1}{2} \times (a + b + c + d) = 105^\circ
\Rightarrow a + b + c + d = 210^\circ \tag{2}
$$

Subtracting equation (2) from equation (1):

$$
(f + e + d + c) - (a + b + c + d) = 220^\circ - 210^\circ
\Rightarrow f + e - a - b = 10^\circ \tag{3}
$$

From equation (2), we can isolate $ c + d $:

$$
c + d = 210^\circ - (a + b) \tag{4}
$$

---

### Step 5: Use the Angle at $ D $ to Find $ \alpha $

The angle at $ D $, $ \alpha $, intercepts arc $ CE $ that does **not** contain $ D $. This arc is the **remaining** part of the circle after subtracting arc $ CD + DE = c + d $:

$$
\text{arc } CE = 360^\circ - (c + d)
$$

So:

$$
\alpha = \frac{1}{2} \times (360^\circ - (c + d)) = 180^\circ - \frac{c + d}{2}
$$

From equation (4), $ c + d = 210^\circ - (a + b) $. But from equation (3), we know:

$$
f + e = a + b + 10^\circ
$$

And from equation (2), $ a + b + c + d = 210^\circ $, so:

$$
c + d = 210^\circ - (a + b)
$$

Substitute into the expression for $ \alpha $:

$$
\alpha = 180^\circ - \frac{210^\circ - (a + b)}{2}
$$

But from equation (3), $ f + e = a + b + 10^\circ $, and from equation (1), $ f + e + c + d = 220^\circ $, so:

$$
a + b + 10^\circ + c + d = 220^\circ
\Rightarrow a + b + c + d = 210^\circ
$$

So, $ c + d = 210^\circ - (a + b) $, and we can now compute:

$$
\alpha = 180^\circ - \frac{210^\circ - (a + b)}{2}
$$

But we don't need to compute $ a + b $ explicitly. Instead, we can use the fact that:

$$
c + d = 70^\circ \quad \text{(from earlier)}
$$

So:

$$
\alpha = 180^\circ - \frac{70^\circ}{2} = 180^\circ - 35^\circ = \boxed{145^\circ}
$$

---

### Final Answer

$$
\boxed{145}
$$

Solution:
Labeling our vertices will help a great deal, as will drawing a few radii: [asy]
pair pA, pB, pC, pD, pE, pF, pO;
pO = (0, 0);
pA = pO + dir(-10);
pB = pO + dir(60);
pC = pO + dir(130);
pD = pO + dir(170);
pE = pO + dir(-160);
pF = pO + dir(-80);
draw(pA--pB--pC--pD--pE--pF--pA);
draw(pA--pO--pC--pO--pE--pO, red);
draw(circle(pO, 1));
label("$O$", pO, NE);
label("$A$", pA, E);
label("$B$", pB, NE);
label("$C$", pC, NW);
label("$D$", pD, W);
label("$E$", pE, SW);
label("$F$", pF, S);
label("$105^\circ$", pF, N * 2);
label("$110^\circ$", pB, SW * 1.5);
label("$\alpha$", pD, E);
[/asy] First of all, we see that $\angle ABC = 110^\circ$ must be half of the major arc ${AEC},$ thus arc ${AEC} = 2 \cdot \angle ABC.$ Then, the minor arc ${AC}$ must be $360^\circ - 2 \cdot \angle ABC = 360^\circ - 2 \cdot 110^\circ = 140^\circ.$

Likewise, the minor arc ${EA}$ must be $360^\circ - 2 \cdot \angle EFA = 360^\circ - 2 \cdot 105^\circ = 150^\circ,$ and the minor arc ${CE}$ is $360^\circ - 2 \alpha.$ Now, arc ${AC},$ ${CE},$ and ${EA}$ must add up to $360^\circ,$ which means that \begin{align*}
360^\circ &= (360^\circ - 2 \alpha) + 140^\circ + 150^\circ\\
360^\circ &= 650^\circ - 2\alpha\\
2\alpha &= 290^\circ\\
\alpha &= \boxed{145^\circ}.
\end{align*}