# Euler's theorem in geometry

Short description: On distance between centers of a triangle
Euler's theorem:
$\displaystyle{ d=|IO| =\sqrt{R (R-2r)} }$

In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by[1][2] $\displaystyle{ d^2=R (R-2r) }$ or equivalently $\displaystyle{ \frac{1}{R-d} + \frac{1}{R+d} = \frac{1}{r}, }$ where $\displaystyle{ R }$ and $\displaystyle{ r }$ denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, who published it in 1765.[3] However, the same result was published earlier by William Chapple in 1746.[4]

From the theorem follows the Euler inequality:[5] $\displaystyle{ R \ge 2r, }$ which holds with equality only in the equilateral case.[6]

## Stronger version of the inequality

A stronger version[6] is $\displaystyle{ \frac{R}{r} \geq \frac{abc+a^3+b^3+c^3}{2abc} \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}-1 \geq \frac{2}{3} \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \geq 2, }$ where $\displaystyle{ a }$, $\displaystyle{ b }$, and $\displaystyle{ c }$ are the side lengths of the triangle.

## Euler's theorem for the escribed circle

If $\displaystyle{ r_a }$ and $\displaystyle{ d_a }$ denote respectively the radius of the escribed circle opposite to the vertex $\displaystyle{ A }$ and the distance between its center and the center of the circumscribed circle, then $\displaystyle{ d_a^2=R(R+2r_a) }$.

## Euler's inequality in absolute geometry

Euler's inequality, in the form stating that, for all triangles inscribed in a given circle, the maximum of the radius of the inscribed circle is reached for the equilateral triangle and only for it, is valid in absolute geometry.[7]