Question 2
Let $SABC$ be a tetrahedron. Triangles $DAB,EBC,FCA$ in the plane of $ABC$ are similar respectively to $SAB,SBC,SCA$. Let $O$ be the circumcenter of $DEF$. Suppose $K$ is the “circumsphere of $SABC$ opposite $O$” (tangent to the planes $SAB,SBC,SCA,ABC$, on the opposite side of $ABC$ from $S$, but on the same side of $SAB$ as $C$, the same side of $SBC$ as $A$, and the same side of $SCA$ as $B$). Prove that $K$ touches the plane $ABC$ at $O$.