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if GL bisects HP then P is the incenter of ABC

Source: 2015 Saudi Arabia IMO TST II p1

July 24, 2020
geometryincenter

Problem Statement

Let ABCABC be an acute-angled triangle inscribed in the circle (O)(O), HH the foot of the altitude of ABCABC at AA and PP a point inside ABCABC lying on the bisector of ∠BAC\angle BAC. The circle of diameter APAP cuts (O)(O) again at GG. Let LL be the projection of PP on AHAH. Prove that if GLGL bisects HPHP then PP is the incenter of the triangle ABCABC.
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