In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by[1][2]
From the theorem follows the Euler inequality:[5]
A stronger version[6] is
If and denote respectively the radius of the escribed circle opposite to the vertex and the distance between its center and the center of the circumscribed circle, then .
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]