Explanation:
Okay, so I need to solve this equation:

$$
\frac{(x+1)(x-3)}{5(x+2)(x-4)} + \frac{(x+3)(x-5)}{9(x+4)(x-6)} - \frac{2(x+5)(x-7)}{13(x+6)(x-8)} = \frac{92}{585}.
$$

Hmm, let me think about how to approach this. The equation has three rational expressions added and subtracted, and they all have different denominators. The first thing that comes to mind is that maybe I can simplify each term individually or find a common denominator. But with three terms, that might get really complicated. Let me check if there's a pattern or maybe some substitution that can make this easier.

Looking at the numerators and denominators:

First term: numerator is (x+1)(x-3), denominator is 5(x+2)(x-4).

Second term: numerator is (x+3)(x-5), denominator is 9(x+4)(x-6).

Third term: numerator is 2(x+5)(x-7), denominator is 13(x+6)(x-8).

Wait, the denominators have denominators with linear factors, and the numerators also have linear factors. Maybe there's a substitution that can make the denominators and numerators relate to each other? Let me check the denominators:

Each denominator is something like (x + a)(x - b), where a and b are numbers. Let me see:

First denominator: (x + 2)(x - 4). Let me note that (x + 2) is x + 2, and x - 4 is x - 4. Similarly for the others. Maybe if I let y = x - something? Or perhaps look for symmetry?

Alternatively, maybe each fraction can be simplified. Let me try expanding the numerators and denominators.

First term numerator: (x + 1)(x - 3) = x² - 3x + x - 3 = x² - 2x - 3.

Denominator: 5(x + 2)(x - 4) = 5(x² - 4x + 2x - 8) = 5(x² - 2x - 8).

So first term is (x² - 2x - 3)/[5(x² - 2x - 8)].

Wait, interesting! The numerator and denominator both have similar quadratic expressions. Let me check:

For the first term: numerator is x² - 2x - 3, denominator is x² - 2x - 8. Let me denote z = x² - 2x. Then numerator is z - 3, denominator is z - 8. So the first term is (z - 3)/[5(z - 8)].

Similarly, maybe check the second term?

Second term numerator: (x + 3)(x - 5) = x² - 5x + 3x - 15 = x² - 2x - 15.

Denominator: 9(x + 4)(x - 6) = 9(x² - 6x + 4x - 24) = 9(x² - 2x - 24).

So second term is (x² - 2x - 15)/[9(x² - 2x - 24)]. Again, if I use z = x² - 2x, then numerator is z - 15, denominator is 9(z - 24). So the second term is (z - 15)/[9(z - 24)].

Third term: numerator is 2(x + 5)(x - 7) = 2(x² - 7x + 5x - 35) = 2(x² - 2x - 35).

Denominator: 13(x + 6)(x - 8) = 13(x² - 8x + 6x - 48) = 13(x² - 2x - 48).

So third term is 2(x² - 2x - 35)/[13(x² - 2x - 48)]. Again, with z = x² - 2x, numerator is 2(z - 35), denominator is 13(z - 48). Therefore, third term is 2(z - 35)/[13(z - 48)].

Wow! So all three terms can be expressed in terms of z = x² - 2x. So maybe substituting z into the equation will simplify things.

So let me write the entire equation in terms of z.

Original equation:

First term: [z - 3]/[5(z - 8)] + [z - 15]/[9(z - 24)] - [2(z - 35)]/[13(z - 48)] = 92/585.

So, now the equation is:

$$
\frac{z - 3}{5(z - 8)} + \frac{z - 15}{9(z - 24)} - \frac{2(z - 35)}{13(z - 48)} = \frac{92}{585}
$$

This seems more manageable. Now, I need to solve for z, then relate back to x.

First, note that z = x² - 2x. Also, need to remember that denominators cannot be zero. So z - 8 ≠ 0, z - 24 ≠ 0, z - 48 ≠ 0. Therefore, z ≠ 8, 24, 48. Also, original denominators have factors (x + 2)(x - 4), etc. So original x cannot be -2, 4, -4, 6, -6, 8. So these x values are excluded. But since z = x² - 2x, let me check if these x values lead to z = 8, 24, 48?

Let me check:

x = -2: z = (-2)^2 - 2*(-2) = 4 + 4 = 8. So z = 8 is excluded.

x = 4: z = 4² - 2*4 = 16 - 8 = 8. So z = 8 is excluded here as well.

Similarly x = -4: z = (-4)^2 - 2*(-4) = 16 + 8 = 24. So z = 24 is excluded.

x = 6: z = 6² - 2*6 = 36 - 12 = 24. So z = 24 is excluded here.

x = -6: z = (-6)^2 - 2*(-6) = 36 + 12 = 48. So z = 48 is excluded.

x = 8: z = 8² - 2*8 = 64 - 16 = 48. So z = 48 is excluded here.

Therefore, the z values are excluded as mentioned. So, when solving for z, we need to avoid z = 8, 24, 48.

So now, the equation is in terms of z. Let me write it again:

$$
\frac{z - 3}{5(z - 8)} + \frac{z - 15}{9(z - 24)} - \frac{2(z - 35)}{13(z - 48)} = \frac{92}{585}
$$

Now, maybe I can combine these fractions. However, the denominators are 5(z - 8), 9(z - 24), 13(z - 48). The denominators are different. But maybe if I find a common denominator? That might be very complicated. Let me check if there's another way.

Alternatively, notice that denominators are multiples of each other? Let me check:

The denominators are 5(z - 8), 9(z - 24), 13(z - 48). Not sure if they have common factors. Let me compute the LCM of denominators. But since they are different linear terms multiplied by constants, the LCM would be 5*9*13*(z - 8)(z - 24)(z - 48). That's going to be a big expression. Maybe not the best way.

Alternatively, perhaps I can make substitution for each term? Let me consider each term separately. Let me think.

Alternatively, maybe there's a pattern in the numerators and denominators. Let me check the constants:

First term has denominator 5, second 9, third 13. Wait, 5, 9, 13... differences of 4. So 5, then 5 + 4 = 9, then 9 + 4 = 13. Interesting. Also, the numerators:

First term numerator: z - 3, which is similar to the denominator z - 8, but shifted by 5. Similarly, second term numerator is z - 15, denominator z - 24, which is shifted by 9. Third term numerator is z - 35, denominator z - 48, shifted by 13. Wait, 8 - 3 = 5, 24 - 15 = 9, 48 - 35 = 13. So each numerator is denominator minus (5,9,13 respectively). Interesting. Maybe that's a clue? Let me check.

Let me think: if I write denominators as D1 = 5(z - 8), D2 = 9(z - 24), D3 = 13(z - 48). The numerators are (z - 3) = D1 numerator, but maybe not. Wait, if I take first term: (z - 3)/[5(z - 8)]. Let me think if there is a relationship between numerator and denominator. For example, z - 8 = denominator's inside, so if I write (z - 8) = denominator's inside, so if I write (z - 3) = (z - 8) + 5. Similarly, (z - 15) = (z - 24) + 9, and (z - 35) = (z - 48) + 13. So:

First term: [(z - 8) + 5]/[5(z - 8)] = [ (z - 8)/5(z - 8) ) + 5/(5(z - 8)) ] = 1/5 + 1/(z - 8). Similarly for the other terms.

Wait, that seems promising. Let me verify:

First term:

(z - 3)/[5(z - 8)] = [ (z - 8) + 5 ] / [5(z - 8) ] = [ (z - 8)/[5(z - 8)] ) + 5/[5(z - 8)] ] = 1/5 + 1/(z - 8).

Similarly, second term:

(z - 15)/[9(z - 24)] = [ (z - 24) + 9 ] / [9(z - 24) ] = [ (z - 24)/[9(z - 24)] ) + 9/[9(z - 24)] ] = 1/9 + 1/(z - 24).

Third term:

- [2(z - 35)]/[13(z - 48)] = -2[ (z - 48) + 13 ] / [13(z - 48) ] = -2[ (z - 48)/[13(z - 48)] + 13/[13(z - 48)] ] = -2[ 1/13 + 1/(z - 48) ] = -2/13 - 2/(z - 48).

Therefore, substituting back into the original equation:

First term + second term - third term:

[1/5 + 1/(z - 8)] + [1/9 + 1/(z - 24)] - [2/13 + 2/(z - 48)] = 92/585.

So simplifying the constants and the fractions:

Constants: 1/5 + 1/9 - 2/13

Fractions: 1/(z - 8) + 1/(z - 24) - 2/(z - 48)

So the entire equation becomes:

[1/5 + 1/9 - 2/13] + [1/(z - 8) + 1/(z - 24) - 2/(z - 48)] = 92/585.

Therefore, if I compute the constants first:

Let me calculate 1/5 + 1/9 - 2/13.

To compute this, find common denominator. Let me compute denominators 5, 9, 13. LCM of 5, 9, 13. Since they are all co-prime, LCM is 5*9*13 = 585. So let's convert each fraction:

1/5 = 117/585 (since 585 / 5 = 117)

1/9 = 65/585 (585 / 9 = 65)

-2/13 = -90/585 (since 585 /13 = 45; 45*2=90, so -2/13 = -90/585)

Therefore, adding them up: 117 + 65 - 90 = (117 + 65) = 182 - 90 = 92. Therefore, 92/585.

So the equation simplifies to:

92/585 + [1/(z - 8) + 1/(z - 24) - 2/(z - 48)] = 92/585.

Subtract 92/585 from both sides:

[1/(z - 8) + 1/(z - 24) - 2/(z - 48)] = 0.

So now the equation reduces to:

1/(z - 8) + 1/(z - 24) - 2/(z - 48) = 0.

This is a much simpler equation. Let me focus on solving this.

So:

1/(z - 8) + 1/(z - 24) = 2/(z - 48)

Let me combine the left side first. Let me compute 1/(z - 8) + 1/(z - 24):

Common denominator is (z - 8)(z - 24). So:

[(z - 24) + (z - 8)] / [(z - 8)(z - 24)] = [2z - 32] / [(z - 8)(z - 24)].

Simplify numerator: 2z - 32 = 2(z - 16).

Therefore, left side becomes 2(z - 16)/[(z - 8)(z - 24)].

So equation:

2(z - 16)/[(z - 8)(z - 24)] = 2/(z - 48).

Divide both sides by 2:

(z - 16)/[(z - 8)(z - 24)] = 1/(z - 48).

Cross-multiplying:

(z - 16)(z - 48) = (z - 8)(z - 24).

Let me expand both sides.

Left side: (z - 16)(z - 48) = z² - 48z - 16z + 16*48 = z² - 64z + 768.

Right side: (z - 8)(z - 24) = z² - 24z - 8z + 192 = z² - 32z + 192.

Set left = right:

z² - 64z + 768 = z² - 32z + 192.

Subtract z² from both sides:

-64z + 768 = -32z + 192.

Bring all terms to left side:

-64z + 768 + 32z - 192 = 0 => (-32z) + 576 = 0 => -32z = -576 => z = (-576)/(-32) = 18.

So z = 18. Therefore, z = 18.

But z was defined as z = x² - 2x. So we have:

x² - 2x = 18 => x² - 2x - 18 = 0.

Solving this quadratic equation. Using quadratic formula:

x = [2 ± sqrt( ( -2 )² - 4*1*(-18) )]/2 = [2 ± sqrt(4 + 72)]/2 = [2 ± sqrt(76)]/2.

Simplify sqrt(76) = sqrt(4*19) = 2*sqrt(19). So:

x = [2 ± 2√19]/2 = 1 ± √19.

So the solutions are x = 1 + √19 and x = 1 - √19.

But wait! I need to check if these solutions are valid, i.e., they don't make any denominators zero in the original equation. Remember that z cannot be 8, 24, 48. Let me check z = 18. 18 is not equal to 8, 24, or 48, so it's okay. Also, need to check if x = 1 + √19 or 1 - √19 are excluded. Let me check:

Original excluded x values are -2, 4, -4, 6, -6, 8.

Compute 1 + √19 ≈ 1 + 4.358 ≈ 5.358, which is not excluded.

Similarly, 1 - √19 ≈ 1 - 4.358 ≈ -3.358, which is not excluded. So both solutions are acceptable.

Therefore, the real roots are 1 + √19 and 1 - √19.

But let me verify these solutions in the original equation to make sure there's no mistake in the algebra.

Let me pick x = 1 + √19.

First, compute z = x² - 2x. Since we found z = 18, and we already confirmed that z is acceptable. So if z = 18, then the equation reduces correctly. Therefore, according to our steps, this should satisfy the equation. But let me check with numbers.

Alternatively, since the steps were algebraic and we checked that z = 18 leads to the solution, and there were no restrictions that exclude these x values, then they should be valid.

But let me test x = 1 + √19 in the original equation. Maybe plug into the equation.

But that might be tedious. Let me try.

First, compute x = 1 + √19. Let me compute z = x² - 2x = 18. So z = 18 is correct. Then, according to the previous steps, the equation reduces to z = 18, which led us to the solutions. Since we did all steps correctly, and cross-verified the constants, I think this is correct. However, to be thorough, maybe check if substituting z = 18 into the equation after simplifying gives 0.

Let me check the equation after simplifying: 1/(z - 8) + 1/(z - 24) - 2/(z - 48) = 0.

With z = 18:

1/(18 - 8) + 1/(18 - 24) - 2/(18 - 48) = 1/10 + 1/(-6) - 2/(-30) = 1/10 - 1/6 + 1/15.

Compute common denominator, let's say 30:

(3/30 - 5/30 + 2/30) = (3 - 5 + 2)/30 = 0/30 = 0. So yes, z = 18 is correct.

Therefore, solutions are correct.

Hence, the real roots are x = 1 ± √19.

Therefore, the answer should be 1 + √19 and 1 - √19.

But let me check if the original equation could have more solutions? Since we had a quadratic in z, which gave one solution z = 18, leading to two solutions for x. Since we didn't have any other solutions, and the equation in z was linear (after cross-multiplying), so only one solution for z, so two solutions for x. Therefore, these are the only real roots.

Therefore, the final answer should be 1 + √19 and 1 - √19.

**Final Answer**
The real roots are \boxed{1 + \sqrt{19}} and \boxed{1 - \sqrt{19}}.

To solve the equation

$$
\frac{(x+1)(x-3)}{5(x+2)(x-4)} + \frac{(x+3)(x-5)}{9(x+4)(x-6)} - \frac{2(x+5)(x-7)}{13(x+6)(x-8)} = \frac{92}{585},
$$

we begin by observing a pattern in the structure of the rational expressions. Each term has a quadratic numerator and a quadratic denominator of similar form but shifted constants. This suggests a substitution that can simplify the problem significantly.

---

### Step 1: Substitution

Let’s define:

$$
z = x^2 - 2x
$$

This substitution simplifies each rational expression because the numerator and denominator of each term will involve expressions in terms of $ z $, where the numerator and denominator differ by constants.

Let’s express each term in terms of $ z $:

- First term:
  $$
  \frac{(x+1)(x-3)}{5(x+2)(x-4)} = \frac{z - 3}{5(z - 8)}
  $$

- Second term:
  $$
  \frac{(x+3)(x-5)}{9(x+4)(x-6)} = \frac{z - 15}{9(z - 24)}
  $$

- Third term:
  $$
  \frac{2(x+5)(x-7)}{13(x+6)(x-8)} = \frac{2(z - 35)}{13(z - 48)}
  $$

Now the equation becomes:

$$
\frac{z - 3}{5(z - 8)} + \frac{z - 15}{9(z - 24)} - \frac{2(z - 35)}{13(z - 48)} = \frac{92}{585}
$$

---

### Step 2: Rewrite Terms

We notice that the numerators and denominators can be related by constants:

- For the first term:
  $$
  (z - 3) = (z - 8) + 5 \Rightarrow \frac{z - 3}{5(z - 8)} = \frac{1}{5} + \frac{1}{z - 8}
  $$

- Second term:
  $$
  \frac{z - 15}{9(z - 24)} = \frac{1}{9} + \frac{1}{z - 24}
  $$

- Third term:
  $$
  \frac{2(z - 35)}{13(z - 48)} = \frac{2}{13} + \frac{2}{z - 48}
  $$

Now substituting into the original equation:

$$
\left( \frac{1}{5} + \frac{1}{z - 8} \right) + \left( \frac{1}{9} + \frac{1}{z - 24} \right) - \left( \frac{2}{13} + \frac{2}{z - 48} \right) = \frac{92}{585}
$$

Group constants and fractions:

- Constants:
  $$
  \frac{1}{5} + \frac{1}{9} - \frac{2}{13}
  $$
  Convert to denominators of 585:

  $$
  \frac{117}{585} + \frac{65}{585} - \frac{90}{585} = \frac{92}{585}
  $$

- Variables:
  $$
  \frac{1}{z - 8} + \frac{1}{z - 24} - \frac{2}{z - 48}
  $$

Now the equation becomes:

$$
\frac{92}{585} + \left( \frac{1}{z - 8} + \frac{1}{z - 24} - \frac{2}{z - 48} \right) = \frac{92}{585}
$$

Subtracting $ \frac{92}{585} $ from both sides:

$$
\frac{1}{z - 8} + \frac{1}{z - 24} - \frac{2}{z - 48} = 0
$$

---

### Step 3: Solve the Equation for $ z $

Group the terms:

$$
\frac{1}{z - 8} + \frac{1}{z - 24} = \frac{2}{z - 48}
$$

Find common denominator for the left-hand side:

$$
\frac{(z - 24) + (z - 8)}{(z - 8)(z - 24)} = \frac{2}{z - 48}
\Rightarrow \frac{2(z - 16)}{(z - 8)(z - 24)} = \frac{2}{z - 48}
$$

Cancel the 2:

$$
\frac{z - 16}{(z - 8)(z - 24)} = \frac{1}{z - 48}
$$

Cross-multiply:

$$
(z - 16)(z - 48) = (z - 8)(z - 24)
$$

Expand both sides:

- Left:
  $$
  z^2 - 48z - 16z + 768 = z^2 - 64z + 768
  $$

- Right:
  $$
  z^2 - 24z - 8z + 192 = z^2 - 32z + 192
  $$

Set equal:

$$
z^2 - 64z + 768 = z^2 - 32z + 192
$$

Subtract $ z^2 $ from both sides:

$$
-64z + 768 = -32z + 192
\Rightarrow -32z = -576
\Rightarrow z = 18
$$

---

### Step 4: Solve for $ x $

Recall that $ z = x^2 - 2x $. So:

$$
x^2 - 2x = 18 \Rightarrow x^2 - 2x - 18 = 0
$$

Solving this quadratic:

$$
x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-18)}}{2} = \frac{2 \pm \sqrt{4 + 72}}{2} = \frac{2 \pm \sqrt{76}}{2}
$$

$$
x = \frac{2 \pm 2\sqrt{19}}{2} = 1 \pm \sqrt{19}
$$

---

### Step 5: Final Check

We check that $ x = 1 \pm \sqrt{19} $ are not excluded values, and that no denominator becomes zero in the original equation. Since the excluded values are:

$$
x = \{-6, -4, -2, 4, 6, 8\}
$$

and $ \sqrt{19} \approx 4.358 $, we have:

- $ 1 + \sqrt{19} \approx 5.358 $
- $ 1 - \sqrt{19} \approx -3.358 $

Neither of these is in the excluded set.

---

### ✅ Final Answer

$$
\boxed{1 + \sqrt{19}}, \boxed{1 - \sqrt{19}}
$$

[Assistant]
Explanation:
Okay, so I need to solve this equation:

$$
\frac{(x+1)(x-3)}{5(x+2)(x-4)} + \frac{(x+3)(x-5)}{9(x+4)(x-6)} - \frac{2(x+5)(x-7)}{13(x+6)(x-8)} = \frac{92}{585}.
$$

Hmm, let me think about how to approach this. The equation has three rational expressions added and subtracted, and they all have different denominators. The first thing that comes to mind is that maybe I can simplify each term individually or find a common denominator. But with three terms, that might get really complicated. Let me check if there's a pattern or maybe some substitution that can make this easier.

Looking at the numerators and denominators:

First term: numerator is (x+1)(x-3), denominator is 5(x+2)(x-4).

Second term: numerator is (x+3)(x-5), denominator is 9(x+4)(x-6).

Third term: numerator is 2(x+5)(x-7), denominator is 13(x+6)(x-8).

Wait, the denominators have denominators with linear factors, and the numerators also have linear factors. Maybe there's a substitution that can make the denominators and numerators relate to each other? Let me check the denominators:

Each denominator is something like (x + a)(x - b), where a and b are numbers. Let me see:

First denominator: (x + 2)(x - 4). Let me note that (x + 2) is x + 2, and x - 4 is x - 4. Similarly for the others. Maybe if I let y = x - something? Or perhaps look for symmetry?

Alternatively, maybe each fraction can be simplified. Let me try expanding the numerators and denominators.

First term numerator: (x + 1)(x - 3) = x² - 3x + x - 3 = x² - 2x - 3.

Denominator: 5(x + 2)(x - 4) = 5(x² - 4x + 2x - 8) = 5(x² - 2x - 8).

So first term is (x² - 2x - 3)/[5(x² - 2x - 8)].

Wait, interesting! The numerator and denominator both have similar quadratic expressions. Let me check:

For the first term: numerator is x² - 2x - 3, denominator is x² - 2x - 8. Let me denote z = x² - 2x. Then numerator is z - 3, denominator is z - 8. So the first term is (z - 3)/[5(z - 8)].

Similarly, maybe check the second term?

Second term numerator: (x + 3)(x - 5) = x² - 5x + 3x - 15 = x² - 2x - 15.

Denominator: 9(x + 4)(x - 6) = 9(x² - 6x + 4x - 24) = 9(x² - 2x - 24).

So second term is (x² - 2x - 15)/[9(x² - 2x - 24)]. Again, if I use z = x² - 2x, then numerator is z - 15, denominator is 9(z - 24). So the second term is (z - 15)/[9(z - 24)].

Third term: numerator is 2(x + 5)(x - 7) = 2(x² - 7x + 5x - 35) = 2(x² - 2x - 35).

Denominator: 13(x + 6)(x - 8) = 13(x² - 8x + 6x - 48) = 13(x² - 2x - 48).

So third term is 2(x² - 2x - 35)/[13(x² - 2x - 48)]. Again, with z = x² - 2x, numerator is 2(z - 35), denominator is 13(z - 48). Therefore, third term is 2(z - 35)/[13(z - 48)].

Wow! So all three terms can be expressed in terms of z = x² - 2x. So maybe substituting z into the equation will simplify things.

So let me write the entire equation in terms of z.

Original equation:

First term: [z - 3]/[5(z - 8)] + [z - 15]/[9(z - 24)] - [2(z - 35)]/[13(z - 48)] = 92/585.

So, now the equation is:

$$
\frac{z - 3}{5(z - 8)} + \frac{z - 15}{9(z - 24)} - \frac{2(z - 35)}{13(z - 48)} = \frac{92}{585}
$$

This seems more manageable. Now, I need to solve for z, then relate back to x.

First, note that z = x² - 2x. Also, need to remember that denominators cannot be zero. So z - 8 ≠ 0, z - 24 ≠ 0, z - 48 ≠ 0. Therefore, z ≠ 8, 24, 48. Also, original denominators have factors (x + 2)(x - 4), etc. So original x cannot be -2, 4, -4, 6, -6, 8. So these x values are excluded. But since z = x² - 2x, let me check if these x values lead to z = 8, 24, 48?

Let me check:

x = -2: z = (-2)^2 - 2*(-2) = 4 + 4 = 8. So z = 8 is excluded.

x = 4: z = 4² - 2*4 = 16 - 8 = 8. So z = 8 is excluded here as well.

Similarly x = -4: z = (-4)^2 - 2*(-4) = 16 + 8 = 24. So z = 24 is excluded.

x = 6: z = 6² - 2*6 = 36 - 12 = 24. So z = 24 is excluded here.

x = -6: z = (-6)^2 - 2*(-6) = 36 + 12 = 48. So z = 48 is excluded.

x = 8: z = 8² - 2*8 = 64 - 16 = 48. So z = 48 is excluded here.

Therefore, the z values are excluded as mentioned. So, when solving for z, we need to avoid z = 8, 24, 48.

So now, the equation is in terms of z. Let me write it again:

$$
\frac{z - 3}{5(z - 8)} + \frac{z - 15}{9(z - 24)} - \frac{2(z - 35)}{13(z - 48)} = \frac{92}{585}
$$

Now, maybe I can combine these fractions. However, the denominators are 5(z - 8), 9(z - 24), 13(z - 48). The denominators are different. But maybe if I find a common denominator? That might be very complicated. Let me check if there's another way.

Alternatively, notice that denominators are multiples of each other? Let me check:

The denominators are 5(z - 8), 9(z - 24), 13(z - 48). Not sure if they have common factors. Let me compute the LCM of denominators. But since they are different linear terms multiplied by constants, the LCM would be 5*9*13*(z - 8)(z - 24)(z - 48). That's going to be a big expression. Maybe not the best way.

Alternatively, perhaps I can make substitution for each term? Let me consider each term separately. Let me think.

Alternatively, maybe there's a pattern in the numerators and denominators. Let me check the constants:

First term has denominator 5, second 9, third 13. Wait, 5, 9, 13... differences of 4. So 5, then 5 + 4 = 9, then 9 + 4 = 13. Interesting. Also, the numerators:

First term numerator: z - 3, which is similar to the denominator z - 8, but shifted by 5. Similarly, second term numerator is z - 15, denominator z - 24, which is shifted by 9. Third term numerator is z - 35, denominator z - 48, shifted by 13. Wait, 8 - 3 = 5, 24 - 15 = 9, 48 - 35 = 13. So each numerator is denominator minus (5,9,13 respectively). Interesting. Maybe that's a clue? Let me check.

Let me think: if I write denominators as D1 = 5(z - 8), D2 = 9(z - 24), D3 = 13(z - 48). The numerators are (z - 3) = D1 numerator, but maybe not. Wait, if I take first term: (z - 3)/[5(z - 8)]. Let me think if there is a relationship between numerator and denominator. For example, z - 8 = denominator's inside, so if I write (z - 8) = denominator's inside, so if I write (z - 3) = (z - 8) + 5. Similarly, (z - 15) = (z - 24) + 9, and (z - 35) = (z - 48) + 13. So:

First term: [(z - 8) + 5]/[5(z - 8)] = [ (z - 8)/5(z - 8) ) + 5/(5(z - 8)) ] = 1/5 + 1/(z - 8). Similarly for the other terms.

Wait, that seems promising. Let me verify:

First term:

(z - 3)/[5(z - 8)] = [ (z - 8) + 5 ] / [5(z - 8) ] = [ (z - 8)/[5(z - 8)] ) + 5/[5(z - 8)] ] = 1/5 + 1/(z - 8).

Similarly, second term:

(z - 15)/[9(z - 24)] = [ (z - 24) + 9 ] / [9(z - 24) ] = [ (z - 24)/[9(z - 24)] ) + 9/[9(z - 24)] ] = 1/9 + 1/(z - 24).

Third term:

- [2(z - 35)]/[13(z - 48)] = -2[ (z - 48) + 13 ] / [13(z - 48) ] = -2[ (z - 48)/[13(z - 48)] + 13/[13(z - 48)] ] = -2[ 1/13 + 1/(z - 48) ] = -2/13 - 2/(z - 48).

Therefore, substituting back into the original equation:

First term + second term - third term:

[1/5 + 1/(z - 8)] + [1/9 + 1/(z - 24)] - [2/13 + 2/(z - 48)] = 92/585.

So simplifying the constants and the fractions:

Constants: 1/5 + 1/9 - 2/13

Fractions: 1/(z - 8) + 1/(z - 24) - 2/(z - 48)

So the entire equation becomes:

[1/5 + 1/9 - 2/13] + [1/(z - 8) + 1/(z - 24) - 2/(z - 48)] = 92/585.

Therefore, if I compute the constants first:

Let me calculate 1/5 + 1/9 - 2/13.

To compute this, find common denominator. Let me compute denominators 5, 9, 13. LCM of 5, 9, 13. Since they are all co-prime, LCM is 5*9*13 = 585. So let's convert each fraction:

1/5 = 117/585 (since 585 / 5 = 117)

1/9 = 65/585 (585 / 9 = 65)

-2/13 = -90/585 (since 585 /13 = 45; 45*2=90, so -2/13 = -90/585)

Therefore, adding them up: 117 + 65 - 90 = (117 + 65) = 182 - 90 = 92. Therefore, 92/585.

So the equation simplifies to:

92/585 + [1/(z - 8) + 1/(z - 24) - 2/(z - 48)] = 92/585.

Subtract 92/585 from both sides:

[1/(z - 8) + 1/(z - 24) - 2/(z - 48)] = 0.

So now the equation reduces to:

1/(z - 8) + 1/(z - 24) - 2/(z - 48) = 0.

This is a much simpler equation. Let me focus on solving this.

So:

1/(z - 8) + 1/(z - 24) = 2/(z - 48)

Let me combine the left side first. Let me compute 1/(z - 8) + 1/(z - 24):

Common denominator is (z - 8)(z - 24). So:

[(z - 24) + (z - 8)] / [(z - 8)(z - 24)] = [2z - 32] / [(z - 8)(z - 24)].

Simplify numerator: 2z - 32 = 2(z - 16).

Therefore, left side becomes 2(z - 16)/[(z - 8)(z - 24)].

So equation:

2(z - 16)/[(z - 8)(z - 24)] = 2/(z - 48).

Divide both sides by 2:

(z - 16)/[(z - 8)(z - 24)] = 1/(z - 48).

Cross-multiplying:

(z - 16)(z - 48) = (z - 8)(z - 24).

Let me expand both sides.

Left side: (z - 16)(z - 48) = z² - 48z - 16z + 16*48 = z² - 64z + 768.

Right side: (z - 8)(z - 24) = z² - 24z - 8z + 192 = z² - 32z + 192.

Set left = right:

z² - 64z + 768 = z² - 32z + 192.

Subtract z² from both sides:

-64z + 768 = -32z + 192.

Bring all terms to left side:

-64z + 768 + 32z - 192 = 0 => (-32z) + 576 = 0 => -32z = -576 => z = (-576)/(-32) = 18.

So z = 18. Therefore, z = 18.

But z was defined as z = x² - 2x. So we have:

x² - 2x = 18 => x² - 2x - 18 = 0.

Solving this quadratic equation. Using quadratic formula:

x = [2 ± sqrt( ( -2 )² - 4*1*(-18) )]/2 = [2 ± sqrt(4 + 72)]/2 = [2 ± sqrt(76)]/2.

Simplify sqrt(76) = sqrt(4*19) = 2*sqrt(19). So:

x = [2 ± 2√19]/2 = 1 ± √19.

So the solutions are x = 1 + √19 and x = 1 - √19.

But wait! I need to check if these solutions are valid, i.e., they don't make any denominators zero in the original equation. Remember that z cannot be 8, 24, 48. Let me check z = 18. 18 is not equal to 8, 24, or 48, so it's okay. Also, need to check if x = 1 + √19 or 1 - √19 are excluded. Let me check:

Original excluded x values are -2, 4, -4, 6, -6, 8.

Compute 1 + √19 ≈ 1 + 4.358 ≈ 5.358, which is not excluded.

Similarly, 1 - √19 ≈ 1 - 4.358 ≈ -3.358, which is not excluded. So both solutions are acceptable.

Therefore, the real roots are 1 + √19 and 1 - √19.

But let me verify these solutions in the original equation to make sure there's no mistake in the algebra.

Let me pick x = 1 + √19.

First, compute z = x² - 2x. Since we found z = 18, and we already confirmed that z is acceptable. So if z = 18, then the equation reduces correctly. Therefore, according to our steps, this should satisfy the equation. But let me check with numbers.

Alternatively, since the steps were algebraic and we checked that z = 18 leads to the solution, and there were no restrictions that exclude these x values, then they should be valid.

But let me test x = 1 + √19 in the original equation. Maybe plug into the equation.

But that might be tedious. Let me try.

First, compute x = 1 + √19. Let me compute z = x² - 2x = 18. So z = 18 is correct. Then, according to the previous steps, the equation reduces to z = 18, which led us to the solutions. Since we did all steps correctly, and cross-verified the constants, I think this is correct. However, to be thorough, maybe check if substituting z = 18 into the equation after simplifying gives 0.

Let me check the equation after simplifying: 1/(z - 8) + 1/(z - 24) - 2/(z - 48) = 0.

With z = 18:

1/(18 - 8) + 1/(18 - 24) - 2/(18 - 48) = 1/10 + 1/(-6) - 2/(-30) = 1/10 - 1/6 + 1/15.

Compute common denominator, let's say 30:

(3/30 - 5/30 + 2/30) = (3 - 5 + 2)/30 = 0/30 = 0. So yes, z = 18 is correct.

Therefore, solutions are correct.

Hence, the real roots are x = 1 ± √19.

Therefore, the answer should be 1 + √19 and 1 - √19.

But let me check if the original equation could have more solutions? Since we had a quadratic in z, which gave one solution z = 18, leading to two solutions for x. Since we didn't have any other solutions, and the equation in z was linear (after cross-multiplying), so only one solution for z, so two solutions for x. Therefore, these are the only real roots.

Therefore, the final answer should be 1 + √19 and 1 - √19.

**Final Answer**
The real roots are \boxed{1 + \sqrt{19}} and \boxed{1 - \sqrt{19}}.

To solve the equation

$$
\frac{(x+1)(x-3)}{5(x+2)(x-4)} + \frac{(x+3)(x-5)}{9(x+4)(x-6)} - \frac{2(x+5)(x-7)}{13(x+6)(x-8)} = \frac{92}{585},
$$

we begin by observing a pattern in the structure of the rational expressions. Each term has a quadratic numerator and a quadratic denominator of similar form but shifted constants. This suggests a substitution that can simplify the problem significantly.

---

### Step 1: Substitution

Let’s define:

$$
z = x^2 - 2x
$$

This substitution simplifies each rational expression because the numerator and denominator of each term will involve expressions in terms of $ z $, where the numerator and denominator differ by constants.

Let’s express each term in terms of $ z $:

- First term:
  $$
  \frac{(x+1)(x-3)}{5(x+2)(x-4)} = \frac{z - 3}{5(z - 8)}
  $$

- Second term:
  $$
  \frac{(x+3)(x-5)}{9(x+4)(x-6)} = \frac{z - 15}{9(z - 24)}
  $$

- Third term:
  $$
  \frac{2(x+5)(x-7)}{13(x+6)(x-8)} = \frac{2(z - 35)}{13(z - 48)}
  $$

Now the equation becomes:

$$
\frac{z - 3}{5(z - 8)} + \frac{z - 15}{9(z - 24)} - \frac{2(z - 35)}{13(z - 48)} = \frac{92}{585}
$$

---

### Step 2: Rewrite Terms

We notice that the numerators and denominators can be related by constants:

- For the first term:
  $$
  (z - 3) = (z - 8) + 5 \Rightarrow \frac{z - 3}{5(z - 8)} = \frac{1}{5} + \frac{1}{z - 8}
  $$

- Second term:
  $$
  \frac{z - 15}{9(z - 24)} = \frac{1}{9} + \frac{1}{z - 24}
  $$

- Third term:
  $$
  \frac{2(z - 35)}{13(z - 48)} = \frac{2}{13} + \frac{2}{z - 48}
  $$

Now substituting into the original equation:

$$
\left( \frac{1}{5} + \frac{1}{z - 8} \right) + \left( \frac{1}{9} + \frac{1}{z - 24} \right) - \left( \frac{2}{13} + \frac{2}{z - 48} \right) = \frac{92}{585}
$$

Group constants and fractions:

- Constants:
  $$
  \frac{1}{5} + \frac{1}{9} - \frac{2}{13}
  $$
  Convert to denominators of 585:

  $$
  \frac{117}{585} + \frac{65}{585} - \frac{90}{585} = \frac{92}{585}
  $$

- Variables:
  $$
  \frac{1}{z - 8} + \frac{1}{z - 24} - \frac{2}{z - 48}
  $$

Now the equation becomes:

$$
\frac{92}{585} + \left( \frac{1}{z - 8} + \frac{1}{z - 24} - \frac{2}{z - 48} \right) = \frac{92}{585}
$$

Subtracting $ \frac{92}{585} $ from both sides:

$$
\frac{1}{z - 8} + \frac{1}{z - 24} - \frac{2}{z - 48} = 0
$$

---

### Step 3: Solve the Equation for $ z $

Group the terms:

$$
\frac{1}{z - 8} + \frac{1}{z - 24} = \frac{2}{z - 48}
$$

Find common denominator for the left-hand side:

$$
\frac{(z - 24) + (z - 8)}{(z - 8)(z - 24)} = \frac{2}{z - 48}
\Rightarrow \frac{2(z - 16)}{(z - 8)(z - 24)} = \frac{2}{z - 48}
$$

Cancel the 2:

$$
\frac{z - 16}{(z - 8)(z - 24)} = \frac{1}{z - 48}
$$

Cross-multiply:

$$
(z - 16)(z - 48) = (z - 8)(z - 24)
$$

Expand both sides:

- Left:
  $$
  z^2 - 48z - 16z + 768 = z^2 - 64z + 768
  $$

- Right:
  $$
  z^2 - 24z - 8z + 192 = z^2 - 32z + 192
  $$

Set equal:

$$
z^2 - 64z + 768 = z^2 - 32z + 192
$$

Subtract $ z^2 $ from both sides:

$$
-64z + 768 = -32z + 192
\Rightarrow -32z = -576
\Rightarrow z = 18
$$

---

### Step 4: Solve for $ x $

Recall that $ z = x^2 - 2x $. So:

$$
x^2 - 2x = 18 \Rightarrow x^2 - 2x - 18 = 0
$$

Solving this quadratic:

$$
x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-18)}}{2} = \frac{2 \pm \sqrt{4 + 72}}{2} = \frac{2 \pm \sqrt{76}}{2}
$$

$$
x = \frac{2 \pm 2\sqrt{19}}{2} = 1 \pm \sqrt{19}
$$

---

### Step 5: Final Check

We check that $ x = 1 \pm \sqrt{19} $ are not excluded values, and that no denominator becomes zero in the original equation. Since the excluded values are:

$$
x = \{-6, -4, -2, 4, 6, 8\}
$$

and $ \sqrt{19} \approx 4.358 $, we have:

- $ 1 + \sqrt{19} \approx 5.358 $
- $ 1 - \sqrt{19} \approx -3.358 $

Neither of these is in the excluded set.

---

### ✅ Final Answer

$$
\boxed{1 + \sqrt{19}}, \boxed{1 - \sqrt{19}}
$$

Solution:
Multiplying out each numerator and denominator, we get
\[\frac{x^2 - 2x - 3}{5(x^2 - 2x - 8)} + \frac{x^2 - 2x - 15}{9(x^2 - 2x - 24)} - \frac{2(x^2 - 2x - 35)}{13(x^2 - 2x - 48)} = \frac{92}{585}.\]We can write this as
\[\frac{(x^2 - 2x - 8) + 5}{5(x^2 - 2x - 8)} + \frac{(x^2 - 2x - 24) + 9}{9(x^2 - 2x - 24)} - \frac{2((x^2 - 2x - 48) + 13)}{13(x^2 - 2x - 48)} = \frac{92}{585}.\]Hence,
\[\frac{1}{5} + \frac{1}{x^2 - 2x - 8} + \frac{1}{9} + \frac{1}{x^2 - 2x - 24} - \frac{2}{13} - \frac{2}{x^2 - 2x - 48} = \frac{92}{585}.\]This simplifies to
\[\frac{1}{x^2 - 2x - 8} + \frac{1}{x^2 - 2x - 24} - \frac{2}{x^2 - 2x - 48} = 0.\]Let $y = x^2 - 2x - 48.$  Then
\[\frac{1}{y + 40} + \frac{1}{y + 24} - \frac{2}{y} = 0.\]Multiplying everything by $y(y + 24)(y + 40),$ we get
\[y(y + 24) + y(y + 40) - 2(y + 24)(y + 40) = 0.\]This simplifies to $64y + 1920 = 0,$ so $y = -30.$  Then $x^2 - 2x - 48 = -30,$ or $x^2 - 2x - 18 = 0.$  By the quadratic formula, $x = \boxed{1 \pm \sqrt{19}}.$  (Since the denominators are nonzero for these values, we know that they are not extraneous.)


Explanation:
Okay, let me try to figure out the measure of angle α in this hexagon inscribed in a circle. So first, I remember that when a polygon is inscribed in a circle, it's called a cyclic polygon. For cyclic polygons, there are some properties about the angles and the arcs they subtend. Maybe I can use those?

The problem gives me a hexagon with some labeled angles. Let me try to visualize the Asymptote figure. There's a circle with center O, and the hexagon is ABCDEF. The points are labeled with directions: A is at -10 degrees, B at 60 degrees, C at 130 degrees, D at 170 degrees, E at -160 degrees, and F at -80 degrees. The angles labeled are 105° at F, 110° at B, and α at D.

Wait, actually, I need to make sure I understand which angles are given. The Asymptote code labels "105°" at point F, "110°" at point B, and "α" at point D. So those are the measures of the internal angles at those vertices? Or are they the angles at the center? Hmm, the labels are placed near the points, so I think they are the internal angles of the hexagon at those vertices. But in cyclic polygons, the internal angles are related to the arcs opposite them.

But let me think again. For cyclic polygons, each internal angle is equal to half the sum of the measures of the arcs intercepted by the two angles opposite to it? Wait, no. Let me recall: in a cyclic quadrilateral, the opposite angles are supplementary. For a cyclic polygon with more sides, the formula might be different. However, maybe I can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc.

So, if I can figure out the measures of the arcs between the vertices, maybe I can find the necessary angles. But since it's a hexagon, there are six arcs. Let me denote them as follows: arc AB, arc BC, arc CD, arc DE, arc EF, arc FA. Each of these arcs corresponds to the central angles between consecutive vertices.

But since the hexagon is cyclic, the sum of all arcs should be 360 degrees. So arc AB + arc BC + arc CD + arc DE + arc EF + arc FA = 360°.

But how do I relate the internal angles of the hexagon to these arcs?

Wait, for a cyclic polygon, each internal angle is equal to half the sum of the measures of the arcs intercepted by the two angles that are opposite to it? Wait, maybe not. Let me think about a cyclic quadrilateral. For a cyclic quadrilateral, each internal angle is half the sum of the measures of the arcs intercepted by the two opposite angles. Wait, no. Actually, in a cyclic quadrilateral, each internal angle is half the measure of the arc opposite to it. Wait, no.

Let me recall: in a cyclic quadrilateral, opposite angles are supplementary. That is, they add up to 180°. But why is that? Because each angle is an inscribed angle that intercepts an arc, and the opposite angle intercepts the remaining arc. Since the total circumference is 360°, the intercepted arcs add up to 360°, so the sum of the angles would be half of that, 180°. So for a cyclic quadrilateral, the sum of opposite angles is 180°.

But for a cyclic hexagon, maybe there's a similar relationship? Not sure. Maybe I need to use the formula for the internal angle of a cyclic polygon. Alternatively, maybe I can break the hexagon into triangles or other shapes and use the properties of inscribed angles.

Alternatively, since we know some angles, maybe we can find the arcs corresponding to those angles, then find the other arcs and then use that to compute α?

Wait, let me think. Let me consider the internal angles of the hexagon. Each internal angle of a cyclic polygon can be related to the arcs. For example, in a cyclic polygon, the internal angle at a vertex is equal to 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 triangle. In a triangle inscribed in a circle, the angle at a vertex is half the measure of the arc opposite to it. For a quadrilateral, each internal angle is half the measure of the arc opposite to it? Wait, no. For a quadrilateral, the angle at a vertex is half the measure of the arc that it intercepts. Wait, if I have a cyclic quadrilateral ABCD, then angle ABC is half the measure of arc ADC. So, angle ABC intercepts arc ADC, right? Because if you have points A, B, C, D on the circle, angle at B intercepts arc ADC. So the measure of angle ABC is half of arc ADC.

Similarly, for a cyclic polygon with more sides, each internal angle is half the measure of the arc opposite to it?

But in a hexagon, it's more complicated. Let me try to think. For example, take vertex B. The internal angle at B is formed by sides AB and BC. This angle is an inscribed angle that intercepts an arc. Which arc?

Wait, the internal angle at vertex B in the hexagon. Let me visualize. The internal angle would be formed by chords AB and BC. The angle between AB and BC is at point B. The intercepted arc would be arc AC. Wait, is that correct?

Wait, in a cyclic polygon, the internal angle at a vertex is equal to half the measure of the arc that is opposite to that angle, meaning the arc that is not containing the vertex. Wait, maybe that's the case for quadrilaterals. For example, in a cyclic quadrilateral, angle B intercepts arc ADC. But for a hexagon, maybe the internal angle at vertex B intercepts the arc that is opposite, but how?

Alternatively, maybe there's a general formula. For a cyclic polygon with n sides, each internal angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides? Wait, maybe not. Let me recall that in any polygon, the sum of the internal angles is (n-2)*180°. For a hexagon, that would be (6-2)*180° = 720°. So the total sum of internal angles is 720°.

But how does that relate to the arcs? If I can find each internal angle in terms of arcs, maybe I can sum them?

Alternatively, since each internal angle is related to the arc opposite to it. Wait, maybe I need to think in terms of central angles.

Each side of the polygon corresponds to an arc. Let me denote the arcs as follows:

Let’s denote arc AB = x, arc BC = y, arc CD = z, arc DE = w, arc EF = v, arc FA = u. Then x + y + z + w + v + u = 360°.

Now, the internal angle at a vertex is related to the arcs. For example, at vertex A, the sides are FA and AB. The internal angle at A would be formed by chords FA and AB. The measure of angle at A is equal to half the measure of the arc opposite to angle A, which would be arc BCDE. Wait, arc BCDE is arc BC + CD + DE. But arc BCDE is y + z + w. Therefore, angle at A is 1/2 (y + z + w). Similarly, angle at B would be 1/2 (z + w + v + u)? Wait, no. Wait, maybe not. Let me think again.

Alternatively, the measure of an internal angle in a cyclic polygon is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, if I have angle at point B, formed by sides AB and BC. The intercepted arc would be the arc that is opposite, which is arc from A to C, not passing through B. Which is arc AC. But arc AC is composed of arcs AB + BC? Wait, no. Wait, arc AC is arc AB + arc BC? Wait, no. If you go from A to C through B, that's arc AB + arc BC. But the arc opposite to angle B would be the arc that's not containing B, which is arc AC (the other arc). So if the circle is divided into arcs AB, BC, CD, DE, EF, FA, then the arc AC not containing B would be from A to C going the other way around the circle, which would be arc AF + FE + ED + DC. Wait, that seems complicated. But maybe there's a better way.

Alternatively, using the formula that in a cyclic polygon, the internal angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but that seems vague.

Wait, let me take an example. Let's think of a triangle. In a triangle inscribed in a circle, each internal angle is half the measure of the arc opposite. So for triangle ABC, angle at A is half arc BC. Similarly, in a quadrilateral, angle at A is half the measure of arc opposite, which is arc BCD or arc BC? Wait, no. For a cyclic quadrilateral, the internal angle at A is half the measure of arc BCD. Wait, that might be. Since angle at A is formed by chords AB and AD, so the intercepted arc is BD? No. Wait, if you have cyclic quadrilateral ABCD, angle at A is between AB and AD, which intercepts arc BD. So the measure of angle A is half of arc BD. Similarly, angle at B is half of arc AC. So in that case, angle A + angle C = 1/2 arc BD + 1/2 arc AC = 1/2 (arc BD + arc AC). But arc BD + arc AC = arc ABCD? Wait, no. Let me think.

In a cyclic quadrilateral, the sum of arc AB + arc BC + arc CD + arc DA = 360°. Then, arc BD is arc BC + arc CD, and arc AC is arc AB + arc BC? Wait, no, arc AC is arc AB + arc BC? If the quadrilateral is labeled in order A, B, C, D, then arc AC would be the arc from A to C passing through B, which is arc AB + arc BC. But the other arc AC would be arc AD + arc DC. However, since angle at A intercepts arc BD, which is arc BC + arc CD. So in that case, angle A is 1/2 (arc BD). Similarly, angle C intercepts arc BD as well? Wait, no. Angle C is formed by chords BC and CD, so it intercepts arc BAD. Which is arc BA + arc AD. So angle C is 1/2 (arc BAD). Since arc BAD + arc BCD = 360°, so angle A + angle C = 1/2 (arc BD + arc BAD) = 1/2 (arc BD + arc BAD). Wait, arc BD is part of arc BCD, and arc BAD is the other part. Maybe this is getting too complicated.

Alternatively, since in a cyclic quadrilateral, angle A + angle C = 180°, because they are supplementary. Which would imply that angle A = 1/2 arc BD and angle C = 1/2 arc BAD. Then angle A + angle C = 1/2 (arc BD + arc BAD) = 1/2 (arc BD + arc BAD). But arc BD + arc BAD is 360°, because arc BD is arc BC + arc CD and arc BAD is arc BA + arc AD. Wait, adding them together would be arc BC + arc CD + arc BA + arc AD. But arc BA is same as arc AB, and arc AD is same as arc DA. But arc BC + arc CD + arc AB + arc DA = arc AB + arc BC + arc CD + arc DA = 360°, which is the entire circle. Therefore angle A + angle C = 1/2 (360°) = 180°, which matches the cyclic quadrilateral property. So that seems to check out.

So, going back, for a cyclic quadrilateral, the internal angles are half the measure of their intercepted arcs. So for a cyclic polygon with more sides, maybe each internal angle is half the measure of the arc that is opposite to it? But in a hexagon, how does that work?

Alternatively, maybe there is a general formula. For any cyclic polygon, the measure of an internal angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, in the quadrilateral example, the internal angle at A is half the measure of arc BD. Arc BD is intercepted by the two sides AB and AD? Wait, but sides AB and AD are adjacent to angle A? No, sides AB and AD are adjacent to angle A. Wait, angle at A is formed by sides AB and AD. So the intercepted arc is BD. So, the measure of angle A is half the measure of arc BD. So in general, for any cyclic polygon, the internal angle at a vertex is equal to half the measure of the arc that is opposite to that angle, which is the arc that does not contain the vertex. So in a hexagon, the internal angle at vertex B would be half the measure of the arc that is opposite to it, which would be the arc from A to C that doesn't contain B. Wait, but the arc from A to C not containing B would have to go the long way around the circle, passing through D, E, F. Let me confirm.

If we have a cyclic hexagon ABCDEF, then the internal angle at vertex B is formed by sides AB and BC. The intercepted arc for this angle would be the arc that is opposite, meaning the arc from A to C that does not pass through B. Since the hexagon is labeled in order, the arc from A to C passing through B is arc AB + arc BC, but the arc from A to C not passing through B would be arc AFE... Wait, maybe arc A to C not through B would be arc AF + FE + ED + DC? Wait, maybe not. Let me think again. If the hexagon is labeled in order A, B, C, D, E, F around the circle, then the arc opposite angle B would be the arc that is not between A and C through B. So the arc from A to C that goes through the other side of the circle. Since the circle is 360°, the arc AC (not passing through B) would be equal to 360° minus the arc AC passing through B.

But the arc AC passing through B is arc AB + arc BC. So the other arc AC would be 360° - (arc AB + arc BC). Therefore, the measure of angle at B is half of that. So angle B = 1/2 [360° - (arc AB + arc BC)].

Similarly, for angle at B, which is given as 110°, then 110° = 1/2 [360° - (arc AB + arc BC)].

Therefore, arc AB + arc BC = 360° - 2*angle B = 360° - 2*110° = 360° - 220° = 140°. So arc AB + arc BC = 140°.

Similarly, maybe we can do the same for angle at F, which is given as 105°. Let me check. Angle at F is formed by sides EF and FA. The intercepted arc opposite to angle F would be the arc from E to A that doesn't pass through F. Since the hexagon is labeled A, B, C, D, E, F, the arc from E to A passing through F is arc EF + arc FA, so the other arc from E to A not passing through F would be arc EDCBA? Wait, no. The arc from E to A not passing through F would be arc ED + DC + CB + BA. So, similar to angle B, angle at F intercepts an arc that is 360° minus the arc EF + arc FA. Therefore, angle F = 1/2 [360° - (arc EF + arc FA)].

Given angle F is 105°, so:

105° = 1/2 [360° - (arc EF + arc FA)] =>

arc EF + arc FA = 360° - 2*105° = 360° - 210° = 150°.

So arc EF + arc FA = 150°.

Similarly, we can maybe find expressions for other angles. However, we are asked for angle α at D. Let me note that angle at D is labeled α. So if I can find the measure of angle D, I can find α. Let's try applying the same logic.

Angle at D is formed by sides CD and DE. The intercepted arc opposite to angle D would be the arc from C to E that doesn't pass through D. The arc from C to E through D is arc CD + arc DE, so the opposite arc would be 360° minus that. Therefore, angle D = 1/2 [360° - (arc CD + arc DE)] = 1/2 [arc FA + arc AB + arc BC + arc EF]. Wait, because arc C to E not passing through D would be arc C to F to E? Wait, if you go from C to E not passing through D, then in the hexagon, the vertices are A, B, C, D, E, F. So going from C to E not passing through D would go through F and A? Wait, maybe. Let me think.

If you have points A, B, C, D, E, F in order around the circle. So from C to E, the arc through D would be C to D to E. The arc not through D would be C to F to E? Wait, but C to F to E is arc CF + arc FE. But arc CF is arc CD + DE + EF? Wait, no. Wait, if going from C to F, since the order is C, D, E, F, so arc from C to F through D, E is CD + DE + EF. But the arc from C to F not passing through D would be arc C to A to B to F? Wait, that seems like a longer way. Wait, maybe I need to think about how the circle is divided.

Alternatively, since angle at D is formed by sides CD and DE, so the intercepted arc opposite is the arc not containing D, which would be arc from C to E that does not include D. So starting at C, going around the circle the other way to E. Since the points are in order A, B, C, D, E, F, going from C to E without passing through D would go through F, A, B? Wait, no. If you go from C to E the other way around the circle, you would go through F, then A, then B, then C? Wait, no, starting at C, going the other direction would pass through D? Wait, confusion arises. Maybe I need to assign specific positions based on the Asymptote code.

Looking back at the Asymptote code:

pA = pO + dir(-10); which is 10 degrees below the positive x-axis.

pB = dir(60): 60 degrees from the positive x-axis.

pC = dir(130): 130 degrees.

pD = dir(170): 170 degrees.

pE = dir(-160): which is equivalent to 20 degrees below the negative x-axis, so -160 is same as 200 degrees.

pF = dir(-80): which is 80 degrees below the positive x-axis, so same as 280 degrees.

Wait, dir(theta) in Asymptote gives a point on the unit circle at angle theta degrees from the positive x-axis, counterclockwise. So let me note the positions:

Point A: -10°, which is 350°, so located in the fourth quadrant.

Point B: 60°, first quadrant.

Point C: 130°, second quadrant.

Point D: 170°, second quadrant, near the top.

Point E: -160°, which is equivalent to 200°, third quadrant.

Point F: -80°, equivalent to 280°, fourth quadrant.

So the order around the circle is A (350°), then B (60°), then C (130°), then D (170°), then E (200°), then F (280°), and back to A. Wait, but 280° is still before 350°, so maybe the order is A (350°), then going counterclockwise: next is B at 60°, but 60° is actually counterclockwise from A (which is at 350°). Wait, no, going counterclockwise from A (350°), the next point would be B at 60°, which is 60° - 350° = -290°, but since it's counterclockwise, actually, 350° to 60° is going counterclockwise by 60° - 350° + 360° = 60° - 350° + 360° = 70°? Wait, maybe I should think in terms of the order in which the points are placed.

Wait, in the Asymptote code, the hexagon is drawn as A-B-C-D-E-F-A. So the order around the circle is A, B, C, D, E, F. Given their angle positions:

- A is at -10° (350°)
- B is at 60°
- C is at 130°
- D is at 170°
- E is at -160° (which is 200°)
- F is at -80° (which is 280°)

So arranging these in order counterclockwise:

Starting from A (350°), moving counterclockwise, the next point is B at 60°, then C at 130°, then D at 170°, then E at 200°, then F at 280°, and back to A at 350°. Wait, but 280° to 350° is 70° counterclockwise. So that seems to make sense.

Therefore, the arcs between the points are:

arc AB: from A (350°) to B (60°). Since going counterclockwise, the arc AB is 60° - 350° + 360° = 60° - 350° + 360° = 70°.

Wait, let me compute the measure of arc AB. If points A and B are at 350° and 60°, respectively, then the arc AB going counterclockwise is 60° - 350° + 360° = 70°, yes. Alternatively, the smaller arc between A and B can be calculated as the difference in their angles. But since 350° to 60° is 70°, and the other arc would be 360° - 70° = 290°, but since we are moving counterclockwise following the order of the hexagon, arc AB is 70°.

Similarly, arc BC: from B (60°) to C (130°), which is 130° - 60° = 70°.

arc CD: from C (130°) to D (170°), which is 170° - 130° = 40°.

arc DE: from D (170°) to E (200°), which is 200° - 170° = 30°.

arc EF: from E (200°) to F (280°), which is 280° - 200° = 80°.

arc FA: from F (280°) back to A (350°), which is 350° - 280° = 70°.

Wait, let me check that. From F at 280° counterclockwise to A at 350°, that's 350° - 280° = 70°, so arc FA is 70°.

So all arcs:

arc AB = 70°

arc BC = 70°

arc CD = 40°

arc DE = 30°

arc EF = 80°

arc FA = 70°

Let me check that they sum to 360°:

70 + 70 + 40 + 30 + 80 + 70 = 70*3 + 70 + 40 + 30 + 80? Wait: 70 +70 is 140, +40 is 180, +30 is 210, +80 is 290, +70 is 360. Yes, perfect. So that's correct.

But wait, according to the Asymptote code, the hexagon is inscribed in a circle with points A to F as above. So, if these arcs are known, then maybe we can compute the internal angles? But wait, in the problem, some angles are given. Let me check if the given angles match with this.

The problem says that angle at B is 110°, and angle at F is 105°, and angle at D is α. Let me check if the angle at B is indeed 110°.

Earlier, we had a formula for angle at vertex: angle = 1/2 [360° - (arc AB + arc BC)]. But if arc AB + arc BC = 70 + 70 = 140°, then angle at B would be 1/2 (360 - 140) = 1/2 * 220° = 110°. Which matches the given value of 110° at B. Similarly, for angle at F, which is given as 105°, let's check.

Angle at F is formed by sides EF and FA. The intercepted arc opposite to angle F is 360° - (arc EF + arc FA). Arc EF + arc FA = 80° + 70° = 150°, so angle at F is 1/2 (360° - 150°) = 1/2 * 210° = 105°, which matches the given value of 105°. Great! So the formulas work.

Therefore, if I want to find angle at D, which is α, I can use the same approach.

Angle D is formed by sides CD and DE. The intercepted arc opposite to angle D is 360° - (arc CD + arc DE). So angle D = 1/2 [360° - (arc CD + arc DE)].

Given arc CD is 40°, arc DE is 30°, so their sum is 40 + 30 = 70°. Therefore, angle D = 1/2 (360 - 70) = 1/2 * 290° = 145°. Therefore, α = 145°?

Wait, but let me check if this is correct. Alternatively, since we have all the arcs, perhaps we can compute internal angles for all vertices.

Let me verify with another angle. For example, let me check angle at A. Using the same formula, angle at A is 1/2 [360° - (arc FA + arc AB)]? Wait, angle at A is formed by sides FA and AB. The intercepted arc opposite to angle A would be arc BCDE. Wait, according to earlier logic, angle at A would be 1/2 [360° - (arc FA + arc AB)]. Since arc FA + arc AB = 70 + 70 = 140°, so angle A = 1/2 (360 - 140) = 110°? Wait, but that would be same as angle at B. But in the diagram, angle at B is given as 110°, so angle at A would also be 110°? Let me check with actual arcs.

Alternatively, since arc opposite to angle A is arc BCDE. Which is arc BC + arc CD + arc DE = 70 + 40 + 30 = 140°, so angle A is 1/2 * 140° = 70°? Wait, conflicting results. So which one is correct? There's an inconsistency here.

Wait, why the discrepancy? When I first thought angle at A is 1/2 [360° - (arc FA + arc AB)] which gave 110°, but if I think angle A intercepts arc BCDE which is 140°, so angle A is half of that 70°, which conflicts. So there's a problem here. Which one is correct?

Let me think again. If angle A is formed by sides FA and AB, then the angle at A intercepts arc opposite to it. So which arc? In cyclic polygons, the measure of an inscribed angle is half the measure of its intercepted arc. So angle at A is an inscribed angle intercepting arc opposite. Which arc?

In a cyclic polygon, the internal angle at a vertex is equal to half the measure of the arc that is subtended by the angle. Wait, if I have angle at A, which is formed by chords AF and AB. The angle between AF and AB. The intercepted arc is the arc that is opposite to angle A, which would be arc that does not include A. Wait, intercepted arc is the arc that is cut off by the two sides of the angle, but not containing the vertex. Wait, in the case of a triangle, the angle intercepts the arc opposite. In a quadrilateral, the angle intercepts the arc opposite. So for angle at A in the hexagon, the intercepted arc is the arc that is opposite, meaning not containing A. Since the sides are AF and AB, the intercepted arc would be from F to B not passing through A. Which is arc FBC, which is arc FB (which is arc FA + arc AB, but going the other way? Wait, confusion again.

Alternatively, perhaps I need to use the formula that in any cyclic polygon, the measure of an internal angle is equal to half the sum of the measure of the arcs intercepted by the two adjacent sides? Wait, no.

Wait, let me try to think of another approach. Since we have all arcs, maybe we can compute the internal angles by dividing the hexagon into triangles from the center, and then using the fact that each internal angle can be related to the central angles. Wait, but how?

Alternatively, for each internal angle of the polygon, the angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. For example, in the triangle, the angle is half the intercepted arc. For quadrilaterals, each internal angle is half the intercepted arc. Wait, for quadrilateral ABCD, angle at A is half arc BCD. If I think in terms of arcs, arc BCD is arc BC + arc CD. So angle at A = 1/2 (arc BC + arc CD). Similarly, angle at C would be 1/2 (arc DA + arc AB). Therefore, in general, for a cyclic polygon, the internal angle at a vertex is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but in a quadrilateral, that would be angle A is half arc BC + arc CD. Which is arc opposite to angle A.

Wait, in the hexagon, taking angle at A, formed by sides FA and AB. The intercepted arcs would be... Wait, FA is from F to A, AB is from A to B. The angle at A is formed by these two sides. The intercepted arc would be the arc that is opposite to angle A, which is arc from F to B not passing through A. Which is arc FBC? Arc from F to B not through A. Since points are ordered A, B, C, D, E, F, going counterclockwise, the arc from F to B not passing through A would be arc F, E, D, C, B. Which is arc FE + arc ED + arc DC + arc CB.

But given that arc FE is 80°, arc ED is 30°, arc DC is 40°, arc CB is 70°, so that arc is 80 + 30 + 40 + 70 = 220°, so angle at A would be half that? Which is 110°. Which matches the previous result that angle A = 1/2 [360 - (arc FA + arc AB)] = 110°. Because arc FA + arc AB is 70 + 70 = 140°, so 360 - 140 = 220°, half is 110°. So same result. So maybe the formula is angle at A is half (360° - arc FA - arc AB). Which is equivalent to half the intercepted arc (arc FBCDE...). So in this case, angle at A is 110°, as computed.

Similarly, angle at D is formed by sides CD and DE. The intercepted arc is arc from C to E not passing through D. Which is arc CFAEB? Let me compute it. Starting at C, going counterclockwise not through D, which would be arc C to F to A to B to E. Which is arc CF + arc FA + arc AB + arc BE? Wait, need to figure out the exact arc. Alternatively, since the intercepted arc for angle D is the arc opposite to angle D, which is arc from C to E not containing D. Given the points are A, B, C, D, E, F in order, arc from C to E not containing D would be going through F, A, B? Wait, from C to E not through D would be going the other way around the circle. From C, going counterclockwise the other way would go through F, E? Wait, no. Wait, if you are at point C and you want to go to E without passing through D, you can go directly from C to E through F? Wait, points are ordered A, B, C, D, E, F. So from C, moving counterclockwise, the next points are D, E, F. If you want to go from C to E without passing through D, you would go through F, then A, B? Wait, this is confusing. Maybe I need to calculate the measure.

Since angle at D is formed by sides CD and DE, so the intercepted arc is the arc that's opposite. Using the formula angle D = 1/2 [360° - (arc CD + arc DE)].

Earlier, we had arc CD = 40°, arc DE = 30°, so their sum is 70°, so angle D = 1/2 (360 - 70) = 1/2 * 290 = 145°, which would be α. Alternatively, if I calculate the intercepted arc, it would be 360 - (arc CD + arc DE) = 290°, and angle D is half of that, 145°. So that would be the measure of angle D, which is α. So is 145° the correct answer? But let me check with another method to confirm.

Alternatively, since we know all arcs, we can compute all internal angles. Let me try that.

For each internal angle:

Angle at A: 1/2 [360 - (arc FA + arc AB)] = 1/2 [360 - 70 -70] = 1/2 * 220 = 110°.

Angle at B: 1/2 [360 - (arc AB + arc BC)] = 1/2 [360 - 70 -70] = same as angle A, 110°. Wait, but in the Asymptote code, angle at B is labeled as 110°, which matches. So that's correct.

Angle at C: formed by sides BC and CD. The intercepted arc opposite is 360 - (arc BC + arc CD) = 360 - 70 -40 = 250°, so angle C = 1/2 * 250 = 125°.

Angle at D: 1/2 [360 - (arc CD + arc DE)] = 1/2 [360 - 40 - 30] = 1/2 * 290 = 145°, which is α.

Angle at E: formed by sides DE and EF. Intercepted arc opposite is 360 - (arc DE + arc EF) = 360 - 30 -80 = 250°, so angle E = 1/2 * 250 = 125°.

Angle at F: formed by sides EF and FA. Intercepted arc opposite is 360 - (arc EF + arc FA) = 360 -80 -70 = 210°, angle F = 1/2 * 210 = 105°, which matches the given value.

So let's check if the sum of all internal angles is 720°, as it should be for a hexagon.

Sum = 110° + 110° + 125° + 145° + 125° + 105° = let's compute:

110 + 110 = 220

220 + 125 = 345

345 + 145 = 490

490 + 125 = 615

615 + 105 = 720. Perfect, that matches. So this seems consistent.

Therefore, the measure of α is 145 degrees. So I think that's the answer.

But let me check once again with another approach. For example, if we use the central angles. Each arc corresponds to a central angle. For example, arc AB is 70°, so central angle AOB is 70°, etc. If we connect all the vertices to the center, we divide the hexagon into six sectors. However, to find internal angles of the hexagon, perhaps we can consider triangles or other polygons.

Alternatively, for each internal angle, we can think of it as the angle between two adjacent sides, which can be calculated by knowing the adjacent arcs. However, since we already used the method of intercepted arcs and verified that all angles sum to 720°, and the given angles match, the measure of α is indeed 145°.

Therefore, I think the answer is 145 degrees.

**Final Answer**
\boxed{145}

To determine the measure of the angle $ \alpha $ at vertex $ D $ in a cyclic hexagon inscribed in a circle, we use properties of cyclic polygons and the relationship between inscribed angles and their intercepted arcs.

---

### **Step 1: Understanding the Geometric Setup**

The hexagon $ ABCDEF $ is inscribed in a circle, meaning all its vertices lie on the circumference of the circle. We are given:
- $ \angle B = 110^\circ $
- $ \angle F = 105^\circ $
- $ \alpha = \angle D $ (to be determined)

We are told to find the measure of angle $ \alpha $, located at vertex $ D $, which is an internal angle of the hexagon.

---

### **Step 2: Properties of Cyclic Polygons**

In a **cyclic polygon**, each **internal angle** at a vertex is equal to **half the measure of the arc intercepted by the two adjacent sides** (the arc **opposite** the angle, not containing the vertex itself).

For angle $ D $, formed by sides $ CD $ and $ DE $, the intercepted arc is the arc that **does not** contain $ D $ and is formed by the remaining points $ A $, $ B $, and $ F $, i.e., the arc $ AFE $.

So, the key relationship is:

$$
\angle D = \frac{1}{2} \times (\text{Measure of arc opposite to } \angle D)
$$

This means:

$$
\angle D = \frac{1}{2} \times (360^\circ - (\text{arc } CD + \text{arc } DE))
$$

---

### **Step 3: Use of Given Arc Measures**

From the Asymptote code and the positions of each point on the circle, we deduce the arcs as follows (in degrees):

- arc $ AB = 70^\circ $
- arc $ BC = 70^\circ $
- arc $ CD = 40^\circ $
- arc $ DE = 30^\circ $
- arc $ EF = 80^\circ $
- arc $ FA = 70^\circ $

Let’s verify that these arcs sum to $ 360^\circ $ (the total circumference of the circle):

$$
70 + 70 + 40 + 30 + 80 + 70 = 360^\circ \quad \text{✔}
$$

Now, calculate the arc opposite to $ \angle D $, which is:

$$
\text{arc } CD + \text{arc } DE = 40^\circ + 30^\circ = 70^\circ
$$

So,

$$
\alpha = \angle D = \frac{1}{2}(360^\circ - 70^\circ) = \frac{1}{2} \times 290^\circ = 145^\circ
$$

---

### **Step 4: Final Check (Consistency with Sum of Interior Angles)**

A hexagon has $ 6 $ sides, so the sum of its interior angles is:

$$
(6 - 2) \times 180^\circ = 720^\circ
$$

If we calculate all the internal angles using the same formula:
- $ \angle A = \frac{1}{2}(360^\circ - 70^\circ - 70^\circ) = \frac{1}{2}(220^\circ) = 110^\circ $
- $ \angle B = \angle A = 110^\circ $
- $ \angle C = \frac{1}{2}(360^\circ - 70^\circ - 40^\circ) = 125^\circ $
- $ \angle D = \alpha = 145^\circ $
- $ \angle E = \frac{1}{2}(360^\circ - 30^\circ - 80^\circ) = 125^\circ $
- $ \angle F = 105^\circ $

Summing them:

$$
110 + 110 + 125 + 145 + 125 + 105 = 720^\circ \quad \text{✔}
$$

All checks are consistent.

---

### **Final Answer**

$$
\boxed{145}
$$

[Assistant]
Explanation:
Okay, let me try to figure out the measure of angle α in this hexagon inscribed in a circle. So first, I remember that when a polygon is inscribed in a circle, it's called a cyclic polygon. For cyclic polygons, there are some properties about the angles and the arcs they subtend. Maybe I can use those?

The problem gives me a hexagon with some labeled angles. Let me try to visualize the Asymptote figure. There's a circle with center O, and the hexagon is ABCDEF. The points are labeled with directions: A is at -10 degrees, B at 60 degrees, C at 130 degrees, D at 170 degrees, E at -160 degrees, and F at -80 degrees. The angles labeled are 105° at F, 110° at B, and α at D.

Wait, actually, I need to make sure I understand which angles are given. The Asymptote code labels "105°" at point F, "110°" at point B, and "α" at point D. So those are the measures of the internal angles at those vertices? Or are they the angles at the center? Hmm, the labels are placed near the points, so I think they are the internal angles of the hexagon at those vertices. But in cyclic polygons, the internal angles are related to the arcs opposite them.

But let me think again. For cyclic polygons, each internal angle is equal to half the sum of the measures of the arcs intercepted by the two angles opposite to it? Wait, no. Let me recall: in a cyclic quadrilateral, the opposite angles are supplementary. For a cyclic polygon with more sides, the formula might be different. However, maybe I can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc.

So, if I can figure out the measures of the arcs between the vertices, maybe I can find the necessary angles. But since it's a hexagon, there are six arcs. Let me denote them as follows: arc AB, arc BC, arc CD, arc DE, arc EF, arc FA. Each of these arcs corresponds to the central angles between consecutive vertices.

But since the hexagon is cyclic, the sum of all arcs should be 360 degrees. So arc AB + arc BC + arc CD + arc DE + arc EF + arc FA = 360°.

But how do I relate the internal angles of the hexagon to these arcs?

Wait, for a cyclic polygon, each internal angle is equal to half the sum of the measures of the arcs intercepted by the two angles that are opposite to it? Wait, maybe not. Let me think about a cyclic quadrilateral. For a cyclic quadrilateral, each internal angle is half the sum of the measures of the arcs intercepted by the two opposite angles. Wait, no. Actually, in a cyclic quadrilateral, each internal angle is half the measure of the arc opposite to it. Wait, no.

Let me recall: in a cyclic quadrilateral, opposite angles are supplementary. That is, they add up to 180°. But why is that? Because each angle is an inscribed angle that intercepts an arc, and the opposite angle intercepts the remaining arc. Since the total circumference is 360°, the intercepted arcs add up to 360°, so the sum of the angles would be half of that, 180°. So for a cyclic quadrilateral, the sum of opposite angles is 180°.

But for a cyclic hexagon, maybe there's a similar relationship? Not sure. Maybe I need to use the formula for the internal angle of a cyclic polygon. Alternatively, maybe I can break the hexagon into triangles or other shapes and use the properties of inscribed angles.

Alternatively, since we know some angles, maybe we can find the arcs corresponding to those angles, then find the other arcs and then use that to compute α?

Wait, let me think. Let me consider the internal angles of the hexagon. Each internal angle of a cyclic polygon can be related to the arcs. For example, in a cyclic polygon, the internal angle at a vertex is equal to 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 triangle. In a triangle inscribed in a circle, the angle at a vertex is half the measure of the arc opposite to it. For a quadrilateral, each internal angle is half the measure of the arc opposite to it? Wait, no. For a quadrilateral, the angle at a vertex is half the measure of the arc that it intercepts. Wait, if I have a cyclic quadrilateral ABCD, then angle ABC is half the measure of arc ADC. So, angle ABC intercepts arc ADC, right? Because if you have points A, B, C, D on the circle, angle at B intercepts arc ADC. So the measure of angle ABC is half of arc ADC.

Similarly, for a cyclic polygon with more sides, each internal angle is half the measure of the arc opposite to it?

But in a hexagon, it's more complicated. Let me try to think. For example, take vertex B. The internal angle at B is formed by sides AB and BC. This angle is an inscribed angle that intercepts an arc. Which arc?

Wait, the internal angle at vertex B in the hexagon. Let me visualize. The internal angle would be formed by chords AB and BC. The angle between AB and BC is at point B. The intercepted arc would be arc AC. Wait, is that correct?

Wait, in a cyclic polygon, the internal angle at a vertex is equal to half the measure of the arc that is opposite to that angle, meaning the arc that is not containing the vertex. Wait, maybe that's the case for quadrilaterals. For example, in a cyclic quadrilateral, angle B intercepts arc ADC. But for a hexagon, maybe the internal angle at vertex B intercepts the arc that is opposite, but how?

Alternatively, maybe there's a general formula. For a cyclic polygon with n sides, each internal angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides? Wait, maybe not. Let me recall that in any polygon, the sum of the internal angles is (n-2)*180°. For a hexagon, that would be (6-2)*180° = 720°. So the total sum of internal angles is 720°.

But how does that relate to the arcs? If I can find each internal angle in terms of arcs, maybe I can sum them?

Alternatively, since each internal angle is related to the arc opposite to it. Wait, maybe I need to think in terms of central angles.

Each side of the polygon corresponds to an arc. Let me denote the arcs as follows:

Let’s denote arc AB = x, arc BC = y, arc CD = z, arc DE = w, arc EF = v, arc FA = u. Then x + y + z + w + v + u = 360°.

Now, the internal angle at a vertex is related to the arcs. For example, at vertex A, the sides are FA and AB. The internal angle at A would be formed by chords FA and AB. The measure of angle at A is equal to half the measure of the arc opposite to angle A, which would be arc BCDE. Wait, arc BCDE is arc BC + CD + DE. But arc BCDE is y + z + w. Therefore, angle at A is 1/2 (y + z + w). Similarly, angle at B would be 1/2 (z + w + v + u)? Wait, no. Wait, maybe not. Let me think again.

Alternatively, the measure of an internal angle in a cyclic polygon is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, if I have angle at point B, formed by sides AB and BC. The intercepted arc would be the arc that is opposite, which is arc from A to C, not passing through B. Which is arc AC. But arc AC is composed of arcs AB + BC? Wait, no. Wait, arc AC is arc AB + arc BC? Wait, no. If you go from A to C through B, that's arc AB + arc BC. But the arc opposite to angle B would be the arc that's not containing B, which is arc AC (the other arc). So if the circle is divided into arcs AB, BC, CD, DE, EF, FA, then the arc AC not containing B would be from A to C going the other way around the circle, which would be arc AF + FE + ED + DC. Wait, that seems complicated. But maybe there's a better way.

Alternatively, using the formula that in a cyclic polygon, the internal angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but that seems vague.

Wait, let me take an example. Let's think of a triangle. In a triangle inscribed in a circle, each internal angle is half the measure of the arc opposite. So for triangle ABC, angle at A is half arc BC. Similarly, in a quadrilateral, angle at A is half the measure of arc opposite, which is arc BCD or arc BC? Wait, no. For a cyclic quadrilateral, the internal angle at A is half the measure of arc BCD. Wait, that might be. Since angle at A is formed by chords AB and AD, so the intercepted arc is BD? No. Wait, if you have cyclic quadrilateral ABCD, angle at A is between AB and AD, which intercepts arc BD. So the measure of angle A is half of arc BD. Similarly, angle at B is half of arc AC. So in that case, angle A + angle C = 1/2 arc BD + 1/2 arc AC = 1/2 (arc BD + arc AC). But arc BD + arc AC = arc ABCD? Wait, no. Let me think.

In a cyclic quadrilateral, the sum of arc AB + arc BC + arc CD + arc DA = 360°. Then, arc BD is arc BC + arc CD, and arc AC is arc AB + arc BC? Wait, no, arc AC is arc AB + arc BC? If the quadrilateral is labeled in order A, B, C, D, then arc AC would be the arc from A to C passing through B, which is arc AB + arc BC. But the other arc AC would be arc AD + arc DC. However, since angle at A intercepts arc BD, which is arc BC + arc CD. So in that case, angle A is 1/2 (arc BD). Similarly, angle C intercepts arc BD as well? Wait, no. Angle C is formed by chords BC and CD, so it intercepts arc BAD. Which is arc BA + arc AD. So angle C is 1/2 (arc BAD). Since arc BAD + arc BCD = 360°, so angle A + angle C = 1/2 (arc BD + arc BAD) = 1/2 (arc BD + arc BAD). Wait, arc BD is part of arc BCD, and arc BAD is the other part. Maybe this is getting too complicated.

Alternatively, since in a cyclic quadrilateral, angle A + angle C = 180°, because they are supplementary. Which would imply that angle A = 1/2 arc BD and angle C = 1/2 arc BAD. Then angle A + angle C = 1/2 (arc BD + arc BAD) = 1/2 (arc BD + arc BAD). But arc BD + arc BAD is 360°, because arc BD is arc BC + arc CD and arc BAD is arc BA + arc AD. Wait, adding them together would be arc BC + arc CD + arc BA + arc AD. But arc BA is same as arc AB, and arc AD is same as arc DA. But arc BC + arc CD + arc AB + arc DA = arc AB + arc BC + arc CD + arc DA = 360°, which is the entire circle. Therefore angle A + angle C = 1/2 (360°) = 180°, which matches the cyclic quadrilateral property. So that seems to check out.

So, going back, for a cyclic quadrilateral, the internal angles are half the measure of their intercepted arcs. So for a cyclic polygon with more sides, maybe each internal angle is half the measure of the arc that is opposite to it? But in a hexagon, how does that work?

Alternatively, maybe there is a general formula. For any cyclic polygon, the measure of an internal angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, in the quadrilateral example, the internal angle at A is half the measure of arc BD. Arc BD is intercepted by the two sides AB and AD? Wait, but sides AB and AD are adjacent to angle A? No, sides AB and AD are adjacent to angle A. Wait, angle at A is formed by sides AB and AD. So the intercepted arc is BD. So, the measure of angle A is half the measure of arc BD. So in general, for any cyclic polygon, the internal angle at a vertex is equal to half the measure of the arc that is opposite to that angle, which is the arc that does not contain the vertex. So in a hexagon, the internal angle at vertex B would be half the measure of the arc that is opposite to it, which would be the arc from A to C that doesn't contain B. Wait, but the arc from A to C not containing B would have to go the long way around the circle, passing through D, E, F. Let me confirm.

If we have a cyclic hexagon ABCDEF, then the internal angle at vertex B is formed by sides AB and BC. The intercepted arc for this angle would be the arc that is opposite, meaning the arc from A to C that does not pass through B. Since the hexagon is labeled in order, the arc from A to C passing through B is arc AB + arc BC, but the arc from A to C not passing through B would be arc AFE... Wait, maybe arc A to C not through B would be arc AF + FE + ED + DC? Wait, maybe not. Let me think again. If the hexagon is labeled in order A, B, C, D, E, F around the circle, then the arc opposite angle B would be the arc that is not between A and C through B. So the arc from A to C that goes through the other side of the circle. Since the circle is 360°, the arc AC (not passing through B) would be equal to 360° minus the arc AC passing through B.

But the arc AC passing through B is arc AB + arc BC. So the other arc AC would be 360° - (arc AB + arc BC). Therefore, the measure of angle at B is half of that. So angle B = 1/2 [360° - (arc AB + arc BC)].

Similarly, for angle at B, which is given as 110°, then 110° = 1/2 [360° - (arc AB + arc BC)].

Therefore, arc AB + arc BC = 360° - 2*angle B = 360° - 2*110° = 360° - 220° = 140°. So arc AB + arc BC = 140°.

Similarly, maybe we can do the same for angle at F, which is given as 105°. Let me check. Angle at F is formed by sides EF and FA. The intercepted arc opposite to angle F would be the arc from E to A that doesn't pass through F. Since the hexagon is labeled A, B, C, D, E, F, the arc from E to A passing through F is arc EF + arc FA, so the other arc from E to A not passing through F would be arc EDCBA? Wait, no. The arc from E to A not passing through F would be arc ED + DC + CB + BA. So, similar to angle B, angle at F intercepts an arc that is 360° minus the arc EF + arc FA. Therefore, angle F = 1/2 [360° - (arc EF + arc FA)].

Given angle F is 105°, so:

105° = 1/2 [360° - (arc EF + arc FA)] =>

arc EF + arc FA = 360° - 2*105° = 360° - 210° = 150°.

So arc EF + arc FA = 150°.

Similarly, we can maybe find expressions for other angles. However, we are asked for angle α at D. Let me note that angle at D is labeled α. So if I can find the measure of angle D, I can find α. Let's try applying the same logic.

Angle at D is formed by sides CD and DE. The intercepted arc opposite to angle D would be the arc from C to E that doesn't pass through D. The arc from C to E through D is arc CD + arc DE, so the opposite arc would be 360° minus that. Therefore, angle D = 1/2 [360° - (arc CD + arc DE)] = 1/2 [arc FA + arc AB + arc BC + arc EF]. Wait, because arc C to E not passing through D would be arc C to F to E? Wait, if you go from C to E not passing through D, then in the hexagon, the vertices are A, B, C, D, E, F. So going from C to E not passing through D would go through F and A? Wait, maybe. Let me think.

If you have points A, B, C, D, E, F in order around the circle. So from C to E, the arc through D would be C to D to E. The arc not through D would be C to F to E? Wait, but C to F to E is arc CF + arc FE. But arc CF is arc CD + DE + EF? Wait, no. Wait, if going from C to F, since the order is C, D, E, F, so arc from C to F through D, E is CD + DE + EF. But the arc from C to F not passing through D would be arc C to A to B to F? Wait, that seems like a longer way. Wait, maybe I need to think about how the circle is divided.

Alternatively, since angle at D is formed by sides CD and DE, so the intercepted arc opposite is the arc not containing D, which would be arc from C to E that does not include D. So starting at C, going around the circle the other way to E. Since the points are in order A, B, C, D, E, F, going from C to E without passing through D would go through F, A, B? Wait, no. If you go from C to E the other way around the circle, you would go through F, then A, then B, then C? Wait, no, starting at C, going the other direction would pass through D? Wait, confusion arises. Maybe I need to assign specific positions based on the Asymptote code.

Looking back at the Asymptote code:

pA = pO + dir(-10); which is 10 degrees below the positive x-axis.

pB = dir(60): 60 degrees from the positive x-axis.

pC = dir(130): 130 degrees.

pD = dir(170): 170 degrees.

pE = dir(-160): which is equivalent to 20 degrees below the negative x-axis, so -160 is same as 200 degrees.

pF = dir(-80): which is 80 degrees below the positive x-axis, so same as 280 degrees.

Wait, dir(theta) in Asymptote gives a point on the unit circle at angle theta degrees from the positive x-axis, counterclockwise. So let me note the positions:

Point A: -10°, which is 350°, so located in the fourth quadrant.

Point B: 60°, first quadrant.

Point C: 130°, second quadrant.

Point D: 170°, second quadrant, near the top.

Point E: -160°, which is equivalent to 200°, third quadrant.

Point F: -80°, equivalent to 280°, fourth quadrant.

So the order around the circle is A (350°), then B (60°), then C (130°), then D (170°), then E (200°), then F (280°), and back to A. Wait, but 280° is still before 350°, so maybe the order is A (350°), then going counterclockwise: next is B at 60°, but 60° is actually counterclockwise from A (which is at 350°). Wait, no, going counterclockwise from A (350°), the next point would be B at 60°, which is 60° - 350° = -290°, but since it's counterclockwise, actually, 350° to 60° is going counterclockwise by 60° - 350° + 360° = 60° - 350° + 360° = 70°? Wait, maybe I should think in terms of the order in which the points are placed.

Wait, in the Asymptote code, the hexagon is drawn as A-B-C-D-E-F-A. So the order around the circle is A, B, C, D, E, F. Given their angle positions:

- A is at -10° (350°)
- B is at 60°
- C is at 130°
- D is at 170°
- E is at -160° (which is 200°)
- F is at -80° (which is 280°)

So arranging these in order counterclockwise:

Starting from A (350°), moving counterclockwise, the next point is B at 60°, then C at 130°, then D at 170°, then E at 200°, then F at 280°, and back to A at 350°. Wait, but 280° to 350° is 70° counterclockwise. So that seems to make sense.

Therefore, the arcs between the points are:

arc AB: from A (350°) to B (60°). Since going counterclockwise, the arc AB is 60° - 350° + 360° = 60° - 350° + 360° = 70°.

Wait, let me compute the measure of arc AB. If points A and B are at 350° and 60°, respectively, then the arc AB going counterclockwise is 60° - 350° + 360° = 70°, yes. Alternatively, the smaller arc between A and B can be calculated as the difference in their angles. But since 350° to 60° is 70°, and the other arc would be 360° - 70° = 290°, but since we are moving counterclockwise following the order of the hexagon, arc AB is 70°.

Similarly, arc BC: from B (60°) to C (130°), which is 130° - 60° = 70°.

arc CD: from C (130°) to D (170°), which is 170° - 130° = 40°.

arc DE: from D (170°) to E (200°), which is 200° - 170° = 30°.

arc EF: from E (200°) to F (280°), which is 280° - 200° = 80°.

arc FA: from F (280°) back to A (350°), which is 350° - 280° = 70°.

Wait, let me check that. From F at 280° counterclockwise to A at 350°, that's 350° - 280° = 70°, so arc FA is 70°.

So all arcs:

arc AB = 70°

arc BC = 70°

arc CD = 40°

arc DE = 30°

arc EF = 80°

arc FA = 70°

Let me check that they sum to 360°:

70 + 70 + 40 + 30 + 80 + 70 = 70*3 + 70 + 40 + 30 + 80? Wait: 70 +70 is 140, +40 is 180, +30 is 210, +80 is 290, +70 is 360. Yes, perfect. So that's correct.

But wait, according to the Asymptote code, the hexagon is inscribed in a circle with points A to F as above. So, if these arcs are known, then maybe we can compute the internal angles? But wait, in the problem, some angles are given. Let me check if the given angles match with this.

The problem says that angle at B is 110°, and angle at F is 105°, and angle at D is α. Let me check if the angle at B is indeed 110°.

Earlier, we had a formula for angle at vertex: angle = 1/2 [360° - (arc AB + arc BC)]. But if arc AB + arc BC = 70 + 70 = 140°, then angle at B would be 1/2 (360 - 140) = 1/2 * 220° = 110°. Which matches the given value of 110° at B. Similarly, for angle at F, which is given as 105°, let's check.

Angle at F is formed by sides EF and FA. The intercepted arc opposite to angle F is 360° - (arc EF + arc FA). Arc EF + arc FA = 80° + 70° = 150°, so angle at F is 1/2 (360° - 150°) = 1/2 * 210° = 105°, which matches the given value of 105°. Great! So the formulas work.

Therefore, if I want to find angle at D, which is α, I can use the same approach.

Angle D is formed by sides CD and DE. The intercepted arc opposite to angle D is 360° - (arc CD + arc DE). So angle D = 1/2 [360° - (arc CD + arc DE)].

Given arc CD is 40°, arc DE is 30°, so their sum is 40 + 30 = 70°. Therefore, angle D = 1/2 (360 - 70) = 1/2 * 290° = 145°. Therefore, α = 145°?

Wait, but let me check if this is correct. Alternatively, since we have all the arcs, perhaps we can compute internal angles for all vertices.

Let me verify with another angle. For example, let me check angle at A. Using the same formula, angle at A is 1/2 [360° - (arc FA + arc AB)]? Wait, angle at A is formed by sides FA and AB. The intercepted arc opposite to angle A would be arc BCDE. Wait, according to earlier logic, angle at A would be 1/2 [360° - (arc FA + arc AB)]. Since arc FA + arc AB = 70 + 70 = 140°, so angle A = 1/2 (360 - 140) = 110°? Wait, but that would be same as angle at B. But in the diagram, angle at B is given as 110°, so angle at A would also be 110°? Let me check with actual arcs.

Alternatively, since arc opposite to angle A is arc BCDE. Which is arc BC + arc CD + arc DE = 70 + 40 + 30 = 140°, so angle A is 1/2 * 140° = 70°? Wait, conflicting results. So which one is correct? There's an inconsistency here.

Wait, why the discrepancy? When I first thought angle at A is 1/2 [360° - (arc FA + arc AB)] which gave 110°, but if I think angle A intercepts arc BCDE which is 140°, so angle A is half of that 70°, which conflicts. So there's a problem here. Which one is correct?

Let me think again. If angle A is formed by sides FA and AB, then the angle at A intercepts arc opposite to it. So which arc? In cyclic polygons, the measure of an inscribed angle is half the measure of its intercepted arc. So angle at A is an inscribed angle intercepting arc opposite. Which arc?

In a cyclic polygon, the internal angle at a vertex is equal to half the measure of the arc that is subtended by the angle. Wait, if I have angle at A, which is formed by chords AF and AB. The angle between AF and AB. The intercepted arc is the arc that is opposite to angle A, which would be arc that does not include A. Wait, intercepted arc is the arc that is cut off by the two sides of the angle, but not containing the vertex. Wait, in the case of a triangle, the angle intercepts the arc opposite. In a quadrilateral, the angle intercepts the arc opposite. So for angle at A in the hexagon, the intercepted arc is the arc that is opposite, meaning not containing A. Since the sides are AF and AB, the intercepted arc would be from F to B not passing through A. Which is arc FBC, which is arc FB (which is arc FA + arc AB, but going the other way? Wait, confusion again.

Alternatively, perhaps I need to use the formula that in any cyclic polygon, the measure of an internal angle is equal to half the sum of the measure of the arcs intercepted by the two adjacent sides? Wait, no.

Wait, let me try to think of another approach. Since we have all arcs, maybe we can compute the internal angles by dividing the hexagon into triangles from the center, and then using the fact that each internal angle can be related to the central angles. Wait, but how?

Alternatively, for each internal angle of the polygon, the angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. For example, in the triangle, the angle is half the intercepted arc. For quadrilaterals, each internal angle is half the intercepted arc. Wait, for quadrilateral ABCD, angle at A is half arc BCD. If I think in terms of arcs, arc BCD is arc BC + arc CD. So angle at A = 1/2 (arc BC + arc CD). Similarly, angle at C would be 1/2 (arc DA + arc AB). Therefore, in general, for a cyclic polygon, the internal angle at a vertex is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but in a quadrilateral, that would be angle A is half arc BC + arc CD. Which is arc opposite to angle A.

Wait, in the hexagon, taking angle at A, formed by sides FA and AB. The intercepted arcs would be... Wait, FA is from F to A, AB is from A to B. The angle at A is formed by these two sides. The intercepted arc would be the arc that is opposite to angle A, which is arc from F to B not passing through A. Which is arc FBC? Arc from F to B not through A. Since points are ordered A, B, C, D, E, F, going counterclockwise, the arc from F to B not passing through A would be arc F, E, D, C, B. Which is arc FE + arc ED + arc DC + arc CB.

But given that arc FE is 80°, arc ED is 30°, arc DC is 40°, arc CB is 70°, so that arc is 80 + 30 + 40 + 70 = 220°, so angle at A would be half that? Which is 110°. Which matches the previous result that angle A = 1/2 [360 - (arc FA + arc AB)] = 110°. Because arc FA + arc AB is 70 + 70 = 140°, so 360 - 140 = 220°, half is 110°. So same result. So maybe the formula is angle at A is half (360° - arc FA - arc AB). Which is equivalent to half the intercepted arc (arc FBCDE...). So in this case, angle at A is 110°, as computed.

Similarly, angle at D is formed by sides CD and DE. The intercepted arc is arc from C to E not passing through D. Which is arc CFAEB? Let me compute it. Starting at C, going counterclockwise not through D, which would be arc C to F to A to B to E. Which is arc CF + arc FA + arc AB + arc BE? Wait, need to figure out the exact arc. Alternatively, since the intercepted arc for angle D is the arc opposite to angle D, which is arc from C to E not containing D. Given the points are A, B, C, D, E, F in order, arc from C to E not containing D would be going through F, A, B? Wait, from C to E not through D would be going the other way around the circle. From C, going counterclockwise the other way would go through F, E? Wait, no. Wait, if you are at point C and you want to go to E without passing through D, you can go directly from C to E through F? Wait, points are ordered A, B, C, D, E, F. So from C, moving counterclockwise, the next points are D, E, F. If you want to go from C to E without passing through D, you would go through F, then A, B? Wait, this is confusing. Maybe I need to calculate the measure.

Since angle at D is formed by sides CD and DE, so the intercepted arc is the arc that's opposite. Using the formula angle D = 1/2 [360° - (arc CD + arc DE)].

Earlier, we had arc CD = 40°, arc DE = 30°, so their sum is 70°, so angle D = 1/2 (360 - 70) = 1/2 * 290 = 145°, which would be α. Alternatively, if I calculate the intercepted arc, it would be 360 - (arc CD + arc DE) = 290°, and angle D is half of that, 145°. So that would be the measure of angle D, which is α. So is 145° the correct answer? But let me check with another method to confirm.

Alternatively, since we know all arcs, we can compute all internal angles. Let me try that.

For each internal angle:

Angle at A: 1/2 [360 - (arc FA + arc AB)] = 1/2 [360 - 70 -70] = 1/2 * 220 = 110°.

Angle at B: 1/2 [360 - (arc AB + arc BC)] = 1/2 [360 - 70 -70] = same as angle A, 110°. Wait, but in the Asymptote code, angle at B is labeled as 110°, which matches. So that's correct.

Angle at C: formed by sides BC and CD. The intercepted arc opposite is 360 - (arc BC + arc CD) = 360 - 70 -40 = 250°, so angle C = 1/2 * 250 = 125°.

Angle at D: 1/2 [360 - (arc CD + arc DE)] = 1/2 [360 - 40 - 30] = 1/2 * 290 = 145°, which is α.

Angle at E: formed by sides DE and EF. Intercepted arc opposite is 360 - (arc DE + arc EF) = 360 - 30 -80 = 250°, so angle E = 1/2 * 250 = 125°.

Angle at F: formed by sides EF and FA. Intercepted arc opposite is 360 - (arc EF + arc FA) = 360 -80 -70 = 210°, angle F = 1/2 * 210 = 105°, which matches the given value.

So let's check if the sum of all internal angles is 720°, as it should be for a hexagon.

Sum = 110° + 110° + 125° + 145° + 125° + 105° = let's compute:

110 + 110 = 220

220 + 125 = 345

345 + 145 = 490

490 + 125 = 615

615 + 105 = 720. Perfect, that matches. So this seems consistent.

Therefore, the measure of α is 145 degrees. So I think that's the answer.

But let me check once again with another approach. For example, if we use the central angles. Each arc corresponds to a central angle. For example, arc AB is 70°, so central angle AOB is 70°, etc. If we connect all the vertices to the center, we divide the hexagon into six sectors. However, to find internal angles of the hexagon, perhaps we can consider triangles or other polygons.

Alternatively, for each internal angle, we can think of it as the angle between two adjacent sides, which can be calculated by knowing the adjacent arcs. However, since we already used the method of intercepted arcs and verified that all angles sum to 720°, and the given angles match, the measure of α is indeed 145°.

Therefore, I think the answer is 145 degrees.

**Final Answer**
\boxed{145}

To determine the measure of the angle $ \alpha $ at vertex $ D $ in a cyclic hexagon inscribed in a circle, we use properties of cyclic polygons and the relationship between inscribed angles and their intercepted arcs.

---

### **Step 1: Understanding the Geometric Setup**

The hexagon $ ABCDEF $ is inscribed in a circle, meaning all its vertices lie on the circumference of the circle. We are given:
- $ \angle B = 110^\circ $
- $ \angle F = 105^\circ $
- $ \alpha = \angle D $ (to be determined)

We are told to find the measure of angle $ \alpha $, located at vertex $ D $, which is an internal angle of the hexagon.

---

### **Step 2: Properties of Cyclic Polygons**

In a **cyclic polygon**, each **internal angle** at a vertex is equal to **half the measure of the arc intercepted by the two adjacent sides** (the arc **opposite** the angle, not containing the vertex itself).

For angle $ D $, formed by sides $ CD $ and $ DE $, the intercepted arc is the arc that **does not** contain $ D $ and is formed by the remaining points $ A $, $ B $, and $ F $, i.e., the arc $ AFE $.

So, the key relationship is:

$$
\angle D = \frac{1}{2} \times (\text{Measure of arc opposite to } \angle D)
$$

This means:

$$
\angle D = \frac{1}{2} \times (360^\circ - (\text{arc } CD + \text{arc } DE))
$$

---

### **Step 3: Use of Given Arc Measures**

From the Asymptote code and the positions of each point on the circle, we deduce the arcs as follows (in degrees):

- arc $ AB = 70^\circ $
- arc $ BC = 70^\circ $
- arc $ CD = 40^\circ $
- arc $ DE = 30^\circ $
- arc $ EF = 80^\circ $
- arc $ FA = 70^\circ $

Let’s verify that these arcs sum to $ 360^\circ $ (the total circumference of the circle):

$$
70 + 70 + 40 + 30 + 80 + 70 = 360^\circ \quad \text{✔}
$$

Now, calculate the arc opposite to $ \angle D $, which is:

$$
\text{arc } CD + \text{arc } DE = 40^\circ + 30^\circ = 70^\circ
$$

So,

$$
\alpha = \angle D = \frac{1}{2}(360^\circ - 70^\circ) = \frac{1}{2} \times 290^\circ = 145^\circ
$$

---

### **Step 4: Final Check (Consistency with Sum of Interior Angles)**

A hexagon has $ 6 $ sides, so the sum of its interior angles is:

$$
(6 - 2) \times 180^\circ = 720^\circ
$$

If we calculate all the internal angles using the same formula:
- $ \angle A = \frac{1}{2}(360^\circ - 70^\circ - 70^\circ) = \frac{1}{2}(220^\circ) = 110^\circ $
- $ \angle B = \angle A = 110^\circ $
- $ \angle C = \frac{1}{2}(360^\circ - 70^\circ - 40^\circ) = 125^\circ $
- $ \angle D = \alpha = 145^\circ $
- $ \angle E = \frac{1}{2}(360^\circ - 30^\circ - 80^\circ) = 125^\circ $
- $ \angle F = 105^\circ $

Summing them:

$$
110 + 110 + 125 + 145 + 125 + 105 = 720^\circ \quad \text{✔}
$$

All checks are consistent.

---

### **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*}