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Tetrahedron ABCD

Source: Romanian TST 1992 - Day 1 - Problem 3

April 9, 2012
geometry3D geometrytetrahedronspheregeometric transformationhomothetyreflection

Problem Statement

Let ABCDABCD be a tetrahedron; B′,C′,D′B', C', D' be the midpoints of the edges AB,AC,ADAB, AC, AD; GA,GB,GC,GDG_A, G_B, G_C, G_D be the barycentres of the triangles BCD,ACD,ABD,ABCBCD, ACD, ABD, ABC, and GG be the barycentre of the tetrahedron. Show that A,G,GB,GC,GDA, G, G_B, G_C, G_D are all on a sphere if and only if A,G,B′,C′,D′A, G, B', C', D' are also on a sphere.
Dan Brânzei
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