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AP x BH = BQ x AH, bisector, altitude, projections related

Source: 2011 Belarus TST 1.2

June 14, 2020
geometryangle bisectoraltituderatio

Problem Statement

Points LL and HH are marked on the sides ABAB of an acute-angled triangle ABC so that CLCL is a bisector and CHCH is an altitude. Let P,QP,Q be the feet of the perpendiculars from LL to ACAC and BCBC respectively. Prove that AP⋅BH=BQ⋅AHAP \cdot BH = BQ \cdot AH.
I. Gorodnin
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