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Prove that the triangle KJM is isosceles

Source: China TST 2011 - Quiz 3 - D2 - P1

May 20, 2011
geometrycircumcircletrigonometrytrapezoidgeometric transformationreflectionparallelogram

Problem Statement

Let HH be the orthocenter of an acute trangle ABCABC with circumcircle Γ\Gamma. Let PP be a point on the arc BCBC (not containing AA) of Γ\Gamma, and let MM be a point on the arc CACA (not containing BB) of Γ\Gamma such that HH lies on the segment PMPM. Let KK be another point on Γ\Gamma such that KMKM is parallel to the Simson line of PP with respect to triangle ABCABC. Let QQ be another point on Γ\Gamma such that PQ∥BCPQ \parallel BC. Segments BCBC and KQKQ intersect at a point JJ. Prove that △KJM\triangle KJM is an isosceles triangle.
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