KöMaL 2009 A485

Let ABCD be a tetrahedron with circumcenter $O$. Suppose that the points $P$, $Q$ and $R$ are interior points of the edges $AB$, $AC$ and $AD$, respectively. Let $K$, $L$, $M$ and $N$ be the centroids of the triangles PQD , PRC , QRB and PQR , respectively. Prove that if the plane PQR is tangent to the sphere KLMN then $OP$=$OQ$=$OR$.