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Sum of ratios of lengths in tetrahedron

Source: Czech and Slovak Olympiad 1971, National Round, Problem 6

July 9, 2024
geometry3D geometrytetrahedronratio

Problem Statement

Let a tetrahedron ABCDABCD and its inner point OO be given. For any edge ee of ABCDABCD consider the segment f(e)f(e) containing OO such that f(e)∥ef(e)\parallel e and the endpoints of f(e)f(e) lie on the faces of the tetrahedron. Show that ∑e edge  f(e) e=3.\sum_{e\text{ edge}}\,\frac{\,f(e)\,}{e}=3.
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