MathDB

Problem 3

Part of 1988 Bulgaria National Olympiad

Problems(1)

tetrahedron inequality, face areas

Source: Bulgaria 1988 P3

6/15/2021
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?
geometry3D geometrytetrahedroninequalitiesgeometric inequality