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collinear centroids of tetrahedra

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1982 p1

September 10, 2024
geometrytetrahedron3D geometrycollinear

Problem Statement

Given a tetrahedron ABCDABCD and inside the tetrahedron points K,L,M,NK, L, M, N that do not lie on a plane. Denote also the centroids of PP, QQ, RR, SS of the tetrahedrons KBCDKBCD, ALCDALCD, ABMDABMD, ABCNABCN do not lie on a plane. Let TT be the centroid of the tetrahedron ABCD, ToT_o be the centroid of the tetrahedron PQRSPQRS and T1T_1 be the centroid of the tetrahedron KLMNKLMN.
a) Prove that the points T,T0,T1T, T_0, T_1 lie in one straight line. b) Determine the ratio ∣T0T∣:∣T0T1∣|T_0T| : |T_0 T_1|.
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