Euler's theorem in geometry

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 $$\frac{R}{r} \geq \frac{abc+a^3+b^3+c^3}{2abc} \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}-1 \geq \frac{2}{3} \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \geq 2,$$ where $$a$$, $$b$$, and $$c$$ are the side lengths 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 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.