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orthocenter of APQ lies on MN, cyclic ABCD, AB = AD, MN = BN + DM

Source: 2011 Grand Duchy of Lithuania, Mathematical Contest p4 (Baltic Way TST)

October 2, 2020
geometryorthocenterequal segmentscyclic quadrilateral

Problem Statement

In the cyclic quadrilateral ABCDABCD with AB=ADAB = AD, points MM and NN lie on the sides CDCD and BCBC respectively so that MN=BN+DMMN = BN + DM. Lines AMAM and ANAN meet the circumcircle of ABCDABCD again at points PP and QQ respectively. Prove that the orthocenter of the triangle APQAPQ lies on the segment MNMN.
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