MathDB

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

Problem Statement