Euler's formula OI² = R² − 2Rr

Dependencies: Intersecting chords: PA·PB = PC·PD (the "power of a point" identity inside a circle, $IA \cdot IM = R^2 - OI^2$), Incenter-excenter lemma (chicken claw) (the trillium / "chicken-foot" identity $|MI| = |MB|$), Incircle exists (the incircle exists + the incenter $I$ has distance $r$ to each side), Central angle is twice the inscribed angle on the same arc (an inscribed angle equals half the central angle), Angle bisector ⇔ equidistant from sides (angle bisector + half-angle property).

Statement

Let $\triangle ABC$ have circumcenter $O$ and incenter $I$, with circumradius $R$ and inradius $r$. Then the distance between the two centers satisfies

In particular, since $OI^2 \ge 0$ we immediately obtain Euler's inequality

with equality if and only if $\triangle ABC$ is equilateral (i.e. $O = I$).