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Tetrahedron, insphere and centroids

Source: Polish MO 2006

April 8, 2006
geometry3D geometrytetrahedronsphereincentergeometric transformationhomothety

Problem Statement

Tetrahedron ABCDABCD in which AB=CDAB=CD is given. Sphere inscribed in it is tangent to faces ABCABC and ABDABD respectively in KK and LL. Prove that if points KK and LL are centroids of faces ABCABC and ABDABD then tetrahedron ABCDABCD is regular.
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