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Parallel lines thru points on the altitudes of a triangle

Source: XVIII Cono Sur Mathematical Olympiad (2007)

August 10, 2011
geometrycircumcirclegeometric transformationreflectionsymmetryprojective geometrypower of a point

Problem Statement

Let ABCABC be an acute triangle with altitudes ADAD, BEBE, CFCF where DD, EE, FF lie on BCBC, ACAC, ABAB, respectively. Let MM be the midpoint of BCBC. The circumcircle of triangle AEFAEF cuts the line AMAM at AA and XX. The line AMAM cuts the line CFCF at YY. Let ZZ be the point of intersection of ADAD and BXBX. Show that the lines YZYZ and BCBC are parallel.
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