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KR is perpendicular to a median in ABC

Source: Sharygin Finals 2022 9.7

August 1, 2022
geometry

Problem Statement

Let HH be the orthocenter of an acute-angled triangle ABCABC. The circumcircle of triangle AHCAHC meets segments ABAB and BCBC at points PP and QQ. Lines PQPQ and ACAC meet at point RR. A point KK lies on the line PHPH in such a way that ∠KAC=90∘\angle KAC = 90^{\circ}. Prove that KRKR is perpendicular to one of the medians of triangle ABCABC.
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