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Problem 3, Iberoamerican Olympiad 2010

Source:

September 24, 2010
geometrycircumcircleincenterprojective geometrygeometry proposed

Problem Statement

The circle Γ \Gamma is inscribed to the scalene triangle ABCABC. Γ \Gamma is tangent to the sides BC,CABC, CA and ABAB at D,ED, E and FF respectively. The line EFEF intersects the line BCBC at GG. The circle of diameter GDGD intersects Γ \Gamma in RR (R≠D R\neq D ). Let PP, QQ (P≠R,Q≠R P\neq R , Q\neq R ) be the intersections of Γ \Gamma with BRBR and CRCR, respectively. The lines BQBQ and CPCP intersects at XX. The circumcircle of CDECDE meets QRQR at MM, and the circumcircle of BDFBDF meet PRPR at NN. Prove that PMPM, QNQN and RXRX are concurrent.
Author: Arnoldo Aguilar, El Salvador
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