Question 4
Let $ABCD$ be a tetrahedron. Let $A'$ be the circumcenter of the face opposite $A$, and define $B',C',D'$ similarly. Let $p_A$ be the plane through $A$ perpendicular to $C'D'$, $p_B$ the plane through $B$ perpendicular to $D'A'$, $p_C$ the plane through $C$ perpendicular to $A'B'$, and $p_D$ the plane through $D$ perpendicular to $B'C'$. Prove that these four planes have a common point. If this common point coincides with the circumcenter of tetrahedron $ABCD$, must the tetrahedron be regular? Explain.