Let ABC be an arbitrary triangle and O, r, and R its circumcenter, inradius, and circumradius respectively.
Let O_A, O_B, and O_C be the midpoints of sides BC, CA, and AB respectively.

For X = A, B, and C let

     SX = |OOX|  if the line segment OOX intersects
                 the interior of triangle ABC,

        = -|OOX| otherwise.

Prove that r + R = S_A + S_B + S_C.