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AI=IE wanted, incenter, circumcircle, arc midpoint, symm.of incenter wrt side

Source: 2018 Croatia MO p7

August 2, 2020
geometryincentercircumcirclearc midpoint

Problem Statement

Given an acute-angled triangle ABCABC in which ∣AB∣<∣AC∣|AB| <|AC|. Point DD is the midpoint of the shorter arc BCBC of its circumcircle. The point II is the center of its incircle, and the point JJ is symmetric point of II wrt line BCBC. The line DJDJ intersects the circumcircle of the triangle ABCABC at the point EE belonging to the arc ABAB. Prove that ∣AI∣=∣IE∣|AI |= |IE|.
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