Question 26
A plane $\alpha$ intersects the edges $AB$, $BC$, $CD$, and $DA$ of a triangular pyramid $ABCD$ at points $K$, $L$, $M$, and $N$, respectively. The dihedral angles $\angle(KLA,KLM)$, $\angle(LMB,LMN)$, $\angle(MNC,MNK)$, and $\angle(NKD,NKL)$ are equal. Prove that the projections of $A,B,C,D$ onto $\alpha$ are concyclic. (By $\angle(PQR,PQS)$ we denote the dihedral angle of tetrahedron $PQRS$ at edge $PQ$.)