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In geometry, Euler's theorem, named after Leonhard Euler, states that the distance d between the circumcentre and incentre of a triangle can be expressed as

\( d^2=R (R-2r) \, \)

where R and r denote the circumradius and inradius respectively (the radii of the above two circles).

From the theorem follows the Euler inequality:

\( R \ge 2r. \)

A figure for following the proof (which also contains the proof here). Made in GeoGebra software.

Let 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 mid-point 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, then 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

angle BIL = angle A / 2 + angle ABC / 2,

angle IBL = angle ABC / 2 + angle CBL = angle ABC / 2 + angle A / 2,

therefore angle BIL = angle 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).
See also

Bicentric quadrilateral#Fuss' theorem and Carlitz' identity for the relation among the same three variables in bicentric quadrilaterals

External links

Euler's theorem on MathWorld

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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