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BMO Shortlist 2021 G7

Source: BMO Shortlist 2021

May 8, 2022
Balkanshortlist2021geometryexcirclesimilar triangles

Problem Statement

Let ABCABC be an acute scalene triangle. Its CC-excircle tangent to the segment ABAB meets ABAB at point MM and the extension of BCBC beyond BB at point NN. Analogously, its BB-excircle tangent to the segment ACAC meets ACAC at point PP and the extension of BCBC beyond CC at point QQ. Denote by A1A_1 the intersection point of the lines MNMN and PQPQ, and let A2A_2 be defined as the point, symmetric to AA with respect to A1A_1. Define the points B2B_2 and C2C_2, analogously. Prove that △ABC\triangle ABC is similar to △A2B2C2\triangle A_2B_2C_2.
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