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Orthocenter moving on line

Source: German TST 2023 AIMO 7, Problem 1

November 2, 2023
geometrycyclic quadrilateral

Problem Statement

Let ABCABC be a acute angled triangle and let AD,BE,CFAD,BE,CF be its altitudes. X≠A,BX \not=A,B and Y≠A,CY \not=A,C lie on sides ABAB and ACAC, respectively, so that ADXYADXY is a cyclic quadrilateral. Let HH be the orthocenter of triangle AXYAXY.
Prove that HH lies on line EFEF.
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