Euler's theorem in geometry explained
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given byor equivalentlywhere
and
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:which holds with equality only in the equilateral case.
Stronger version of the inequality
A stronger version iswhere
,
, and
are the side lengths of the triangle.
Euler's theorem for the escribed circle
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 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.
See also