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mininal sum of distances from a point to vertices of a tetrahedron

Source: Germany Federal - Bundeswettbewerb Mathematik 2011, round 2, p4

April 14, 2020
3D geometryminSumdistancestetrahedrongeometrygeometric inequality

Problem Statement

Let ABCDABCD be a tetrahedron that is not degenerate and not necessarily regular, where sides ADAD and BCBC have the same length aa, sides BDBD and ACAC have the same length bb, side ABAB has length c1c_1 and the side CDCD has length c2c_2. There is a point PP for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities a,b,c1a, b, c_1 and c2c_2.
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