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Two isogonals in incenter diagram

Source: MEMO 2015, problem T-6

August 28, 2015
geometryincenter

Problem Statement

Let II be the incentre of triangle ABCABC with AB>ACAB>AC and let the line AIAI intersect the side BCBC at DD. Suppose that point PP lies on the segment BCBC and satisfies PI=PDPI=PD. Further, let JJ be the point obtained by reflecting II over the perpendicular bisector of BCBC, and let QQ be the other intersection of the circumcircles of the triangles ABCABC and APDAPD. Prove that ∠BAQ=∠CAJ\angle BAQ=\angle CAJ.
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