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an unusual way to describe incenter of convex polygon

Source: TOT 401 1994 Spring O J3 - Tournament of Towns

June 12, 2024
geometryincentergeometric inequalityangles

Problem Statement

Let OO be a point inside a convex polygon A1A2...AnA_1A_2... A_n such that ∠OA1An≤∠OA1A2,∠OA2A1≤∠OA2A3,...,∠OAn−1An−2≤∠OAn−1An,∠OAnAn−1≤∠OAnA1\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1 and all of these angles are acute. Prove that OO is the centre of the circle inscribed in the polygon.
(V Proizvolov)
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