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incircle, excircle tangent projective geo

Source: Sharygin 2023 - P22 (Grade-10-11)

March 4, 2023
geometrytangentincircleprojective geometrySharygin Geometry OlympiadSharygin 2023anant mudgal geo

Problem Statement

Let ABCABC be a scalene triangle, MM be the midpoint of BC,PBC,P be the common point of AMAM and the incircle of ABCABC closest to AA, and QQ be the common point of the ray AMAM and the excircle farthest from AA. The tangent to the incircle at PP meets BCBC at point XX, and the tangent to the excircle at QQ meets BCBC at YY. Prove that MX=MYMX=MY.
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