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perpendicular wanted, incenter, midpoints related

Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p6

October 16, 2020
geometryincenterperpendicularmidpoints

Problem Statement

Let A1,B1,C1A_1,B_1,C_1 are the midpoints of the sides of the triangle ABC,IABC, I is the center of the circle inscribed in it. Let C2C_2 be the intersection point of lines C1IC_1 I and A1B1A_1B_1. Let C3C_3 be the intersection point of lines CC2CC_2 and ABAB. Prove that line IC3IC_3 is perpendicular to line ABAB.
(A. Zaslavsky)
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