Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$, and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$, respectively. Prove that $$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$where $V$ is the volume of $ABCD$. When does equality hold?
Problem
Source: Bulgaria 1988 P3
Tags: geometry, 3D geometry, tetrahedron, inequalities, geometric inequality