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Concurrency in two circles

Source: Mexican Mathematical Olympiad 2014 Problem 3

November 11, 2014
geometryparallelogramgeometric transformationhomothetygeometry proposed

Problem Statement

Let Γ1\Gamma_1 be a circle and PP a point outside of Γ1\Gamma_1. The tangents from PP to Γ1\Gamma_1 touch the circle at AA and BB. Let MM be the midpoint of PAPA and Γ2\Gamma_2 the circle through PP, AA and BB. Line BMBM cuts Γ2\Gamma_2 at CC, line CACA cuts Γ1\Gamma_1 at DD, segment DBDB cuts Γ2\Gamma_2 at EE and line PEPE cuts Γ1\Gamma_1 at FF, with EE in segment PFPF. Prove lines AFAF, BPBP, and CECE are concurrent.
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