Projections of Midpoints onto Circumradii
Let ABC be an acute triangle with circumcenter O, circumradius , and inradius . Let A', B', and C' be the midpoints of BC, AC, and AB, respectively. Let D, D' be the projections of A' onto the circumradii OB and OC. Define E, E' and F, F' similarly for the points B' and C'. Then .
See problem 62 in N. Altshiller-Court, College Geometry, 2nd ed., Mineola, NY: Dover, 2007 p. 119.