GeoProof — Euler's Inequality explorer

OI² = R² − 2Rr ⟹ R ≥ 2r

Euler's Inequality

Every triangle has a circumcircle (through the vertices, center O, radius R) and an incircle (tangent to the sides, center I, radius r). Euler found a remarkably simple link between the distance OI and the two radii:

OI² = R² − 2Rr

Since a squared length can never be negative, R² − 2Rr ≥ 0, and dividing by R gives the famous inequality:

R ≥ 2r

The circumradius is always at least twice the inradius.

When is it equality?

R = 2r exactly when OI = 0, i.e. when the circumcenter and incenter coincide. That happens only for the equilateral triangle — the single most symmetric case. Drag the corners toward an equilateral shape and watch O and I merge as the 2r bar grows to meet the R bar.

Reading the picture

O is the purple center (equidistant from the vertices); I is the green center (equidistant from the sides). The red segment between them is OI. The bars at the bottom compare R with 2r — the gap between them is exactly what OI measures, through OI² = R² − 2Rr.

Distinct from the Euler line

The Euler line is about the circumcenter, centroid, and orthocenter being collinear. This is a different Euler result — a metric relation between O, I, and the two radii.

A bit of history

Leonhard Euler published this distance formula in 1765. The inequality R ≥ 2r is one of the most quoted facts in triangle geometry.