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K, L, M are collinear iff X is cirumcentre of EOD

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December 31, 2011
geometrycircumcirclegeometry unsolved

Problem Statement

Let HH be the orthocentre and OO be the circumcentre of an acute triangle ABCABC. Let ADAD and BEBE be the altitudes of the triangle with DD on BCBC and EE on CACA. Let K=OD∩BE,L=OE∩ADK =OD \cap BE, L = OE \cap AD. Let XX be the second point of intersection of the circumcircles of triangles HKDHKD and HLEHLE, and let MM be the midpoint of side ABAB. Prove that points K,L,MK, L, M are collinear if and only if XX is the circumcentre of triangle EODEOD.
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