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 byTemplate:R/superscriptTemplate:R/superscript $$d^2=R(R-2r)$$ 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.Template:R/superscript However, the same result was published earlier by William Chapple in 1746.Template:R/superscript

From the theorem follows the Euler inequality:Template:R/superscript $$R\ge 2r,$$ which holds with equality only in the equilateral case.Template:R/superscript

Stronger version of the inequality

A stronger versionTemplate:R/superscript 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 escribed 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.Template:R/superscript

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