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Where are the Circles?

Source: EGMO 2022/1

April 9, 2022
geometryEGMOAngle ChaseEGMO2022collinearincenterAngle Chasing

Problem Statement

Let ABCABC be an acute-angled triangle in which BC<ABBC<AB and BC<CABC<CA. Let point PP lie on segment ABAB and point QQ lie on segment ACAC such that P≠BP \neq B, Q≠CQ \neq C and BQ=BC=CPBQ = BC = CP. Let TT be the circumcenter of triangle APQAPQ, HH the orthocenter of triangle ABCABC, and SS the point of intersection of the lines BQBQ and CPCP. Prove that TT, HH, and SS are collinear.
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