Euler's triangle theorem

Let O and I be the circumcenter and incenter of a triangle with circumradius R and inradius r. Let d be the distance between O and I. Then d^2 = R(R - 2r) (Mackay 1886-1887; Casey 1888, pp. 74-75; Johnson 1929, pp. 186-187; Altshiller-Court 1952, p. 85). This is the simplest case of Poncelet's porism, and is sometimes also known as Euler's triangle theorem. From this theorem, the inequality R>=2r, sometimes known as Euler's inequality, follows immediately.

circumcenter | circumradius | Euler's inequality | incenter | inradius | Poncelet's porism