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r+KX+LY+MZ=2R, with inradius, circumradius and midpoints of sides and arcs

Source: Austrian-Polish 1998

May 23, 2019
geometryinradiuscircumcirclearc midpointmidpoints

Problem Statement

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