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Concept
Euler's theorem in geometry
Summary
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by
or equivalently
where 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.
A stronger version is
where , , and are the side lengths of the triangle.
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.
Official source
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