Let ABC be an acute triangle and let M, N, and P be on CB, AC, and AB, respectively, such that AB=AN+PB, BC=BP+MC, CA=CM+AN. Let ℓ be a line in a different half plane than C with respect to to the line AB such that if X,Y are the projections of A,B on ℓ respectively, AX=AP and BY=BP. Prove that NXYM is a cyclic quadrilateral. cyclic quadrilateralConcyclicequal segments