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circumcircle of HAB passes through orthocenter of HAC

Source: 2016 Saudi Arabia IMO TST , level 4+, II p1

July 27, 2020
geometryorthocentercircumcircle

Problem Statement

Let ABCABC be a triangle inscribed in the circle (O)(O). The bisector of ∠BAC\angle BAC cuts the circle (O)(O) again at DD. Let DEDE be the diameter of (O)(O). Let GG be a point on arc ABAB which does not contain CC. The lines GDGD and BCBC intersect at FF. Let HH be a point on the line AGAG such that FH∥AEFH \parallel AE. Prove that the circumcircle of triangle HABHAB passes through the orthocenter of triangle HACHAC.
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