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P is orthocenter of acute ABC, 3 equal chords given

Source: Tournament of Towns, Senior A-Level , Fall 2019 p2

April 18, 2020
orthocentergeometryChordsequal segments

Problem Statement

Let ABCABC be an acute triangle. Suppose the points A′,B′,C′A',B',C' lie on its sides BC,AC,ABBC,AC,AB respectively and the segments AA′,BB′,CC′AA',BB',CC' intersect in a common point PP inside the triangle. For each of those segments let us consider the circle such that the segment is its diameter, and the chord of this circle that contains the point PP and is perpendicular to this diameter. All three these chords occurred to have the same length. Prove that PP is the orthocenter of the triangle ABCABC.
(Grigory Galperin)
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