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perpendicularity wanted, incenter, centroid, touchpoints with incircle, CA'= AB

Source: 2003 Oral Moscow Geometry Olympiad grade 9 p4

October 26, 2020
geometryincenterCentroidperpendicular

Problem Statement

In triangle ABCABC, MM is the point of intersection of the medians, OO is the center of the inscribed circle, A′,B′,C′A', B', C' are the touchpoints with the sides BC,CA,ABBC, CA, AB, respectively. Prove that if CA′=ABCA'= AB, then OMOM and ABAB are perpendicular.
PS. There is a a typo
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