# Euler's theorem in geometry Euler's theorem:
$d=|IO|={\sqrt {R(R-2r)}}$ In geometry, Euler's theorem states that the distance d between the circumcentre and incentre of a triangle is given by

$d^{2}=R(R-2r)$ or equivalently

${\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},$ where R and 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. However, the same result was published earlier by William Chapple in 1746.

From the theorem follows the Euler inequality:

$R\geq 2r,$ which holds with equality only in the equilateral case.:p. 198

## Proof

Letting O be the circumcentre of triangle ABC, and I be its incentre, the extension of AI intersects the circumcircle at L. Then L is the midpoint of arc BC. Join LO and extend it so that it intersects the circumcircle at M. From I construct a perpendicular to AB, and let D be its foot, so ID = r. It is not difficult to prove that triangle ADI is similar to triangle MBL, so ID / BL = AI / ML, i.e. ID × ML = AI × BL. Therefore 2Rr = AI × BL. Join BI. Because

BIL = ∠ A / 2 + ∠ ABC / 2,
IBL = ∠ ABC / 2 + ∠ CBL = ∠ ABC / 2 + ∠ A / 2,

we have ∠ BIL = ∠ IBL, so BL = IL, and AI × IL = 2Rr. Extend OI so that it intersects the circumcircle at P and Q; then PI × QI = AI × IL = 2Rr, so (R + d)(R − d) = 2Rr, i.e. d2 = R(R − 2r).

## Stronger version of the inequality

A stronger version:p. 198 is

${\frac {R}{r}}\geq {\frac {abc+a^{3}+b^{3}+c^{3}}{2abc}}\geq {\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}-1\geq {\frac {2}{3}}\left({\frac {a}{b}}+{\frac {b}{c}}+{\frac {c}{a}}\right)\geq 2,$

where a, b, c are the sidelengths of the triangle.

## Euler's theorem for the escribed circle

If $r_{a}$  and $d_{a}$  denote respectively the radius of the escribed circle opposite to the vertex $A$  and the distance between its centre and the centre of the circumscribed circle, then $d_{a}^{2}=R(R+2r_{a})$ .

## 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.