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EMC 2017 Harder Geo

Source: EMC 2017

December 26, 2017
homothetyPascal's Theoremcyclic quadrilateraltangent circlesgeometry

Problem Statement

Let ABCABC be a scalene triangle and let its incircle touch sides BCBC, CACA and ABAB at points DD, EE and FF respectively. Let line ADAD intersect this incircle at point XX. Point MM is chosen on the line FXFX so that the quadrilateral AFEMAFEM is cyclic. Let lines AMAM and DEDE intersect at point LL and let QQ be the midpoint of segment AEAE. Point TT is given on the line LQLQ such that the quadrilateral ALDTALDT is cyclic. Let SS be a point such that the quadrilateral TFSATFSA is a parallelogram, and let NN be the second point of intersection of the circumcircle of triangle ASXASX and the line TSTS. Prove that the circumcircles of triangles TANTAN and LSALSA are tangent to each other.
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