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 radii of the circumscribed circle and inscribed circle respectively). The theorem is named for Leonhard Euler, who published it in 1767. 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.

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 (R + d)(R − d) = 2Rr, i.e. d² = R(R − 2r).

A stronger version is

where a, b, c are the sidelengths of the triangle.

If ${\displaystyle r_{a}}$ and ${\displaystyle d_{a}}$ denote respectively the radius of the exscribed circle opposite to the vertex ${\displaystyle A}$ and the distance between its centre and the centre of the circumscribed circle, then ${\displaystyle d_{a}^{2}=R(R+2r_{a})}$.

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