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In geometry, **Euler's theorem** states that the distance *d* between the circumcenter and incenter of a triangle is given by^{[1]}^{[2]}
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.^{[3]} However, the same result was published earlier by William Chapple in 1746.^{[4]}

From the theorem follows the **Euler inequality**:^{[5]}
which holds with equality only in the equilateral case.^{[6]}

A stronger version^{[6]} 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.^{[7]}

- Fuss' theorem for the relation among the same three variables in bicentric quadrilaterals
- Poncelet's closure theorem, showing that there is an infinity of triangles with the same two circles (and therefore the same
*R*,*r*, and*d*) - Egan conjecture, generalization to higher dimensions
- List of triangle inequalities

**^**Johnson, Roger A. (2007) [1929],*Advanced Euclidean Geometry*, Dover Publ., p. 186**^**Dunham, William (2007),*The Genius of Euler: Reflections on his Life and Work*, Spectrum Series, vol. 2, Mathematical Association of America, p. 300, ISBN 9780883855584**^**Leversha, Gerry; Smith, G. C. (November 2007), "Euler and triangle geometry",*The Mathematical Gazette*,**91**(522): 436–452, doi:10.1017/S0025557200182087, JSTOR 40378417, S2CID 125341434**^**Chapple, William (1746), "An essay on the properties of triangles inscribed in and circumscribed about two given circles",*Miscellanea Curiosa Mathematica*,**4**: 117–124. The formula for the distance is near the bottom of p.123.**^**Alsina, Claudi; Nelsen, Roger (2009),*When Less is More: Visualizing Basic Inequalities*, Dolciani Mathematical Expositions, vol. 36, Mathematical Association of America, p. 56, ISBN 9780883853429- ^
^{a}^{b}Svrtan, Dragutin; Veljan, Darko (2012), "Non-Euclidean versions of some classical triangle inequalities",*Forum Geometricorum*,**12**: 197–209; see p. 198 **^**Pambuccian, Victor; Schacht, Celia (2018), "Euler's inequality in absolute geometry",*Journal of Geometry*, 109 (Art. 8): 1–11, doi:10.1007/s00022-018-0414-6, S2CID 125459983

Wikimedia Commons has media related to Euler's theorem in geometry.

- Weisstein, Eric W., "Euler Triangle Formula",
*MathWorld*