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BC//EF wanted, 3 circles related

Source: P3 Francophone Math Olympiad Senior 2022

May 23, 2022
geometrycirclesparallel

Problem Statement

Let ABCABC be a triangle and Γ\Gamma its circumcircle. Denote Δ\Delta the tangent at AA to the circle Γ\Gamma. Γ1\Gamma_1 is a circle tangent to the lines Δ\Delta, (AB)(AB) and (BC)(BC), and EE its touchpoint with the line (AB)(AB). Let Γ2\Gamma_2 be a circle tangent to the lines Δ\Delta, (AC)(AC) and (BC)(BC), and FF its touchpoint with the line (AC)(AC). We suppose that EE and FF belong respectively to the segments [AB][AB] and [AC][AC], and that the two circles Γ1\Gamma_1 and Γ2\Gamma_2 lie outside triangle ABCABC. Show that the lines (BC)(BC) and (EF)(EF) are parallel.
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