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Cyclic implies parallel?

Source: Own. Malaysian IMO TST 2024 P1

April 21, 2024
geometry

Problem Statement

Let ABCABC be an acute triangle with orthocenter HH, and let BEBE and CFCF be the altitudes of the triangle. Choose two points PP and QQ on rays BHBH and CHCH respectively, such that:
∙\bullet PQPQ is parallel to BCBC;
∙\bullet The quadrilateral APHQAPHQ is cyclic.
Suppose the circumcircles of triangles APFAPF and AQEAQE meet again at X≠AX\neq A. Prove that AXAX is parallel to BCBC.
Proposed by Ivan Chan Kai Chin
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