Euler's theorem in geometry

File:Euler theorem2.svg

Euler's theorem:
$d=|IO|=\sqrt{R(R-2r)}$

In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by $$d^2=R(R-2r)$$ 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: $$R\ge 2r,$$ which holds with equality only in the equilateral case.

Stronger version of the inequality

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

Euler's theorem for the excribed circle

If $r_a$ and $d_a$ denote respectively the radius of the excribed circle opposite to the vertex $A$ and the distance between its center and the center 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.

See also

• Fuss' theorem for the relation among the same three variables in bicentric quadrilaterals
• Poncelet's closure theorem, showing that there is an infinity of triangles with the same two circles (and therefore the same R, r, and d)
• Egan conjecture, generalization to higher dimensions
• List of triangle inequalities