Question 51
A plane $\alpha$ intersects edges $AB,BC,CD,DA$ of a tetrahedron $ABCD$ at $K,L,M,N$, respectively. The dihedral angles $\angle(KLA,KLM)$, $\angle(LMB,LMN)$, $\angle(MNC,MNK)$, and $\angle(NKD,NKL)$ are equal. Here $\angle(PQR,PQS)$ denotes the dihedral angle along edge $PQ$ in tetrahedron $PQRS$. Prove that the orthogonal projections of $A,B,C,D$ onto $\alpha$ lie on a circle.