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Ellipsoid problem

Source: 2022 Israel Olympic Revenge P4

July 18, 2022
geometryolympic revenge3D geometrytetrahedron

Problem Statement

A (not necessarily regular) tetrahedron A1A2A3A4A_1A_2A_3A_4 is given in space. For each pair of indices 1≤i<j≤41\leq i<j\leq 4, an ellipsoid with foci Ai,AjA_i,A_j and string length ℓij\ell_{ij}, for positive numbers ℓij\ell_{ij}, is given (in all 6 ellipsoids were built).
For each i=1,2i=1,2, a pair of points Xi≠Xi′X_i\neq X'_i was chosen so that Xi,Xi′X_i, X'_i both belong to all three ellipsoids with AiA_i as one of their foci. Prove that the lines X1X1′,X2X2′X_1X'_1, X_2X'_2 share a point in space if and only if ℓ13+ℓ24=ℓ14+ℓ23\ell_{13}+\ell_{24}=\ell_{14}+\ell_{23} Remark: An ellipsoid with foci P,QP,Q and string length ℓ>∣PQ∣\ell>|PQ| is defined here as the set of points XX in space for which ∣XQ∣+∣XP∣=ℓ|XQ|+|XP|=\ell.
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