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MP = MQ wanted, midpoint of AH, circumcenter, orthocenter

Source: 2011 Saudi Arabia Pre-TST February 4.4 https://artofproblemsolving.com/community/c2745403_2011_

January 1, 2022
geometryequal segmentsorthocenter

Problem Statement

In a triangle ABCABC, let OO be the circumcenter, HH the ortho­center, and MM the midpoint of the segment AHAH. The perpendicular at MM onto OMOM intersects lines ABAB and ACAC at PP and QQ, respectively. Prove that MP=MQMP = MQ.
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