MathDB

Let

A_1A_2... A_{2n + 1}

be a convex polygon,

a_1 = A_1A_2

a_2 ​​= A_2A_3

...

a_{2n} = A_{2n}A_{2n + 1}

a_{2n + 1} = A_{2n + 1}A_1

. Denote by:

\alpha_i = \angle A_i

1 \le i \le 2n + 1

\alpha_{k + 2n + 1} = \alpha_k

k \ge 1

\beta_i = \alpha_{i + 2} + \alpha_{i + 4} +... + \alpha_{i + 2n}

1 \le i \le 2n + 1

. Prove what if

\frac{\alpha_1}{\sin \beta_1}=\frac{\alpha_2}{\sin \beta_2}=...=\frac{\alpha_{2n+1}}{\sin \beta_{2n+1}}

then a circle can be circumscribed around this polygon. Does the inverse statement hold a place?

Problem Statement