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common interior point for all triangles by symmetrics of a point wrt sidelengths

Source: 2021 Czech and Slovak Olympiad III A p6

June 6, 2021
geometryFixed pointfixed

Problem Statement

An acute triangle ABCABC is given. Let us denote XX for each of its inner points Xa,Xb,XcX_a, X_b, X_c its images in axial symmetries sequentially along the lines BC,CA,ABBC, CA, AB. Prove that all XaXbXcX_aX_bX_c triangles have a common interior point.
(Josef Tkadlec)
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