# Euler's theorem (excircles)

## Description

The circumscribed circle or circumcircle of a triangle is a circle which passes through all the vertices of the triangle. The center of this circle is called the circumcenter and its radius is called the circumradius.

An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle’s sides. The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two.

Euler’s theorem states that the distance d between the excircles centrum and circumcenter of a triangle can be expressed by the radius of one of the excircles and the circumradius.

## Variables

R | Circumradius (m) |

r_{ex} | The radius of one of the excircles (m) |

d | The distance between the circumcenter and this excircle's center (m) |