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III Lusophon Mathematical Olympiad 2013 - Problem 6

Source:

August 12, 2013
geometrygeometric transformationhomothety

Problem Statement

Consider a triangle ABCABC. Let SS be a circumference in the interior of the triangle that is tangent to the sides BCBC, CACA, ABAB at the points DD, EE, FF respectively. In the exterior of the triangle we draw three circumferences SAS_A, SBS_B, SCS_C. The circumference SAS_A is tangent to BCBC at LL and to the prolongation of the lines ABAB, ACAC at the points MM, NN respectively. The circumference SBS_B is tangent to ACAC at EE and to the prolongation of the line BCBC at PP. The circumference SCS_C is tangent to ABAB at FF and to the prolongation of the line BCBC at QQ. Show that the lines EPEP, FQFQ and ALAL meet at a point of the circumference SS.
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