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Line XY touches the excircle opposite to A

Source: XVII Sharygin Correspondence Round, P14

March 2, 2021
geometryexcircle

Problem Statement

Let γA,γB,γC\gamma_A, \gamma_B, \gamma_C be excircles of triangle ABCABC, touching the sides BCBC, CACA, ABAB respectively. Let lAl_A denote the common external tangent to γB\gamma_B and γC\gamma_C distinct from BCBC. Define lB,lCl_B, l_C similarly. The tangent from a point PP of lAl_A to γB\gamma_B distinct from lAl_A meets lCl_C at point XX. Similarly the tangent from PP to γC\gamma_C meets lBl_B at YY. Prove that XYXY touches γA\gamma_A.
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