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min |AX|+|BX|+|CX|+|DX| for tetrahedron ABCD

Source: Czech and Slovak Olympiad 1988, National Round, Problem 3

September 13, 2024
geometry3D geometrytetrahedrongeometric inequality

Problem Statement

Given a tetrahedron ABCDABCD with edges ∣AD∣=∣BC∣=a|AD|=|BC|= a, ∣AC∣=∣BD∣=b|AC|=|BD|=b, AB=cAB=c and ∣CD∣=d|CD| = d. Determine the smallest value of the sum ∣AX∣+∣BX∣+∣CX∣+∣DX∣|AX|+|BX|+|CX|+|DX|, where XX is any point in space.
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