Euler's relation: $OI^2$ = $R^2$ - 2Rr (O = circumcenter, I = incenter). Distance from circumcenter to excenters: $OI_a^2$ = $R^2$ + 2$Rr_a$. The nine-point circle has radius R/2 and passes through the midpoints of the sides, feet of altitudes, and midpoints of segments from vertices to orthocenter.