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# Euler's theorem in geometry

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][6]

${\displaystyle R\geq 2r,}$
which holds with equality only in the equilateral case.[7]

## Stronger version of the inequality

A stronger version[7] 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.[8]