Euler's theorem in geometry


In geometry, Euler's theorem states that the distance d between the circumcentre and incentre of a triangle is given by
or equivalently
where R and r denote the circumradius and inradius 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.

Proof

Letting O be the circumcentre of triangle ABC, and I be its incentre, the extension of AI intersects the circumcircle at L. Then L is the midpoint of arc BC. Join LO and extend it so that it intersects the circumcircle at M. From I construct a perpendicular to AB, and let D be its foot, so ID = r. It is not difficult to prove that triangle ADI is similar to triangle MBL, so ID / BL = AI / ML, i.e. ID × ML = AI × BL. Therefore 2Rr = AI × BL. Join BI. Because
we have ∠ BIL = ∠ IBL, so BL = IL, and AI × IL = 2Rr. Extend OI so that it intersects the circumcircle at P and Q; then PI × QI = AI × IL = 2Rr, so = 2Rr, i.e. d2 = R.

Stronger version of the inequality

A stronger version is
where a, b, c are the sidelengths 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 centre and the centre 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.