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Triangle inscribed in another triangle

Source: Cono Sur 2010 #5

November 17, 2015
cono surgeometry proposedgeometry

Problem Statement

The incircle of triangle ABCABC touches sides BCBC, ACAC, and ABAB at D,ED, E, and FF respectively. Let ωa,ωb\omega_a, \omega_b and ωc\omega_c be the circumcircles of triangles EAF,DBFEAF, DBF, and DCEDCE, respectively. The lines DEDE and DFDF cut ωa\omega_a at Ea≠EE_a\neq{E} and Fa≠FF_a\neq{F}, respectively. Let rAr_A be the line EaFaE_{a}F_a. Let rBr_B and rCr_C be defined analogously. Show that the lines rAr_A, rBr_B, and rCr_C determine a triangle with its vertices on the sides of triangle ABCABC.
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