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The Feuerbach point of ABC

Source: 239 2009 S7

July 29, 2020
geometry

Problem Statement

The Feuerbach point (the tangent point of the inscribed circle and the nine-point circle of triangle ABCABC) FF is marked in triangle ABCABC. A1A_1 is on the side BCBC such that AA1AA_1 is the altitude of triangle ABCABC. Prove that the line symmetric to FA1FA_1 with respect to BCBC is perpendicular to IOIO, where OO is the center of the circumcircle of the triangle ABCABC and II is the center of its incircle.
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