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Equal angles iff ratio of lengths are equal

Source: Bulgarian TST 2020 P1

January 23, 2021
geometrycircumcircle

Problem Statement

In acute triangle △ABC\triangle ABC, BC>ACBC>AC, Γ\Gamma is its circumcircle, DD is a point on segment ACAC and EE is the intersection of the circle with diameter CDCD and Γ\Gamma. MM is the midpoint of ABAB and CMCM meets Γ\Gamma again at QQ. The tangents to Γ\Gamma at A,BA,B meet at PP, and HH is the foot of perpendicular from PP to BQBQ. KK is a point on line HQHQ such that QQ lies between HH and KK. Prove that ∠HKP=∠ACE\angle HKP=\angle ACE if and only if KQQH=CDDA\frac{KQ}{QH}=\frac{CD}{DA}.
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