Question 20
Let $ A_1A_2\ldots A_n $ be a regular n-gon. The points $ B_1,\ldots,B_{n-1} $ are defined as follows:
(i) If i = 1 or i = n - 1, then $B_{i}$ is the midpoint of the side $A_{i}A_{i+1}$.
(ii) If $ i \neq 1 $, $ i \neq n-1 $, and S is the intersection point of $ A_1A_{i+1} $ and $ A_nA_i $, then $ B_i $ is the intersection point of the bisector of the angle $ A_iSA_{i+1} $ with $ A_iA_{i+1} $.
Prove the equality
$$ \angle A_{1}B_{1}A_{n}+\angle A_{1}B_{2}A_{n}+\cdots+\angle A_{1}B_{n-1}A_{n}=180^{\circ}. $$
(i) If i = 1 or i = n - 1, then $B_{i}$ is the midpoint of the side $A_{i}A_{i+1}$.
(ii) If $ i \neq 1 $, $ i \neq n-1 $, and S is the intersection point of $ A_1A_{i+1} $ and $ A_nA_i $, then $ B_i $ is the intersection point of the bisector of the angle $ A_iSA_{i+1} $ with $ A_iA_{i+1} $.
Prove the equality
$$ \angle A_{1}B_{1}A_{n}+\angle A_{1}B_{2}A_{n}+\cdots+\angle A_{1}B_{n-1}A_{n}=180^{\circ}. $$