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Excircles Config Geo

Source: Bulgaria National Olympiad 2023 P2

April 8, 2023
geometrycircumcircleangle bisector

Problem Statement

Let ABCABC be an acute triangle and A1,B1,C1A_{1}, B_{1}, C_{1} be the touchpoints of the excircles with the segments BC,CA,ABBC, CA, AB respectively. Let OA,OB,OCO_{A}, O_{B}, O_{C} be the circumcenters of △AB1C1,△BC1A1,△CA1B1\triangle AB_{1}C_{1}, \triangle BC_{1}A_{1}, \triangle CA_{1}B_{1} respectively. Prove that the lines through OA,OB,OCO_{A}, O_{B}, O_{C} respectively parallel to the internal angle bisectors of ∠A,∠B,∠C\angle A,\angle B, \angle C are concurrent.
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