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Incenter of a weird triangle

Source: Baltic Way 2023/12

November 11, 2023
geometry

Problem Statement

Let ABCABC be an acute triangle with AB>ACAB>AC. The internal angle bisector of ∠BAC\angle BAC meets BCBC at DD. Let OO be the circumcenter of ABCABC and let AOAO meet BCBC at EE. Let JJ be the incenter of triangle AEDAED. Show that if ∠ADO=45∘\angle ADO=45^{\circ}, then OJ=JDOJ=JD.
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