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incenter lies on circumircle so the other incenter is collinear with points

Source: KJMO 2010 p3

May 3, 2019
geometryincentercircumcirclecollinearcollinearity

Problem Statement

In an acute triangle △ABC\triangle ABC, let there be point DD on segment AC,EAC, E on segment ABAB such that ∠ADE=∠ABC\angle ADE = \angle ABC. Let the bisector of ∠A\angle A hit BCBC at KK. Let the foot of the perpendicular from KK to DEDE be PP, and the foot of the perpendicular from AA to DEDE be LL. Let QQ be the midpoint of ALAL. If the incenter of △ABC\triangle ABC lies on the circumcircle of △ADE\triangle ADE, prove that P,QP,Q and the incenter of △ADE\triangle ADE are collinear.
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