Question 2
Let $O$ be the center of the circumsphere of tetrahedron $ABCD$. Let $A',B',C',D'$ lie on that sphere so that $AA',BB',CC',DD'$ are diameters. Let $A''$ be the centroid of triangle $BCD$, and define $B'',C'',D''$ analogously. Prove that the lines $A'A'',B'B'',C'C'',D'D''$ are concurrent. Suppose they meet at $X$. Prove that the line through $X$ and the midpoint of $AB$ is perpendicular to $CD$.