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 by¹² $${\displaystyle d^{2}=R(R-2r)}$$ 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 $${\displaystyle R\geq 2r,}$$ which holds with equality only in the equilateral case.⁶

A stronger version⁶ 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.

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 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.⁷