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connecting points with centroids of faces of a polyhedron, in a sequence

Source: Iranian Geometry Olympiad 2018 IGO Intermediate p4

September 19, 2018
geometry3-Dimensional Geometry3D geometrypolyhedronCentroid

Problem Statement

We have a polyhedron all faces of which are triangle. Let PP be an arbitrary point on one of the edges of this polyhedron such that PP is not the midpoint or endpoint of this edge. Assume that P0=PP_0 = P. In each step, connect PiP_i to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point Pi+1P_{i+1}. Continue this process with Pi+1P_{i+1} and the other face containing Pi+1P_{i+1}. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.)
Proposed by Mahdi Etesamifard - Morteza Saghafian
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