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Three incenters and a pair of perpendicular lines

Source: MEMO 2024 T6

August 27, 2024
geometry proposedgeometryincenter

Problem Statement

Let ABCABC be an acute triangle. Let MM be the midpoint of the segment BCBC. Let I,J,KI, J, K be the incenters of triangles ABCABC, ABMABM, ACMACM, respectively. Let P,QP, Q be points on the lines MKMK, MJMJ, respectively, such that ∠AJP=∠ABC\angle AJP=\angle ABC and ∠AKQ=∠BCA\angle AKQ=\angle BCA. Let RR be the intersection of the lines CPCP and BQBQ. Prove that the lines IRIR and BCBC are perpendicular.
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