CZ-SK-MO 1996 Problem 6

Let $K,L,M$ be points on sides $AB,BC,CA$ , respectively, of a triangle $ABC$ such that $AK/AB = BL/BC = CM/CA = 1/3$ . Show that if the circumcircles of the triangles $AKM, BLK, CML$ are equal, then so are the incircles of these triangles.