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Proving three lines concurrent

Source: Own. Malaysian SST 2024 P8

September 5, 2024
geometry

Problem Statement

Given a triangle ABCABC, let II be the incenter, and JJ be the AA-excenter. A line ℓ\ell through AA perpendicular to BCBC intersect the lines BIBI, CICI, BJBJ, CJCJ at PP, QQ, RR, SS respectively. Suppose the angle bisector of ∠BAC\angle BAC meet BCBC at KK, and LL is a point such that ALAL is a diameter in (ABC)(ABC).
Prove that the line KLKL, ℓ\ell, and the line through the centers of circles (IPQ)(IPQ) and (JRS)(JRS), are concurrent.
Proposed by Chuah Jia Herng & Ivan Chan Kai Chin
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