ISL 2004 G5

Let $A_1A_2A_3\ldots A_n$ be a regular $n$ -gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$ . Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$ . Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$ , and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$ .

Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$ .

Proposed by Dusan Dukic, Serbia and Montenegro