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tetrahedron inequality, face areas

Source: Bulgaria 1988 P3

June 15, 2021
geometry3D geometrytetrahedroninequalitiesgeometric inequality

Problem Statement

Let MM be an arbitrary interior point of a tetrahedron ABCDABCD, and let SA,SB,SC,SDS_A,S_B,S_C,S_D be the areas of the faces BCD,ACD,ABD,ABCBCD,ACD,ABD,ABC, respectively. Prove that SA⋅MA+SB⋅MB+SC⋅MC+SD⋅MD≥9V,S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,where VV is the volume of ABCDABCD. When does equality hold?
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