AUSTRIAN-POLISH 1998 Problem 9

Given a triangle $ABC$ , points $K,L,M$ are the midpoints of the sides $BC,CA,AB$ , and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$ .