# Euler's theorem in geometry

__: On distance between centers of a triangle__

**Short description**In geometry, **Euler's theorem** states that the distance *d* between the circumcenter and incenter of a triangle is given by^{[1]}^{[2]}
[math]\displaystyle{ d^2=R (R-2r) }[/math]
or equivalently
[math]\displaystyle{ \frac{1}{R-d} + \frac{1}{R+d} = \frac{1}{r}, }[/math]
where [math]\displaystyle{ R }[/math] and [math]\displaystyle{ r }[/math] 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]}
[math]\displaystyle{ R \ge 2r, }[/math]
which holds with equality only in the equilateral case.^{[6]}

## Stronger version of the inequality

A stronger version^{[6]} is
[math]\displaystyle{ \frac{R}{r} \geq \frac{abc+a^3+b^3+c^3}{2abc} \geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}-1 \geq \frac{2}{3} \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \right) \geq 2, }[/math]
where [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], and [math]\displaystyle{ c }[/math] are the side lengths of the triangle.

## Euler's theorem for the escribed circle

If [math]\displaystyle{ r_a }[/math] and [math]\displaystyle{ d_a }[/math] denote respectively the radius of the escribed circle opposite to the vertex [math]\displaystyle{ A }[/math] and the distance between its center and the center of the circumscribed circle, then [math]\displaystyle{ d_a^2=R(R+2r_a) }[/math].

## 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.^{[7]}

## See also

- 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*) - List of triangle inequalities

## References

- ↑ Johnson, Roger A. (2007),
*Advanced Euclidean Geometry*, Dover Publ., p. 186 - ↑ Dunham, William (2007),
*The Genius of Euler: Reflections on his Life and Work*, Spectrum Series,**2**, Mathematical Association of America, p. 300, ISBN 9780883855584, https://books.google.com/books?id=M4-zUnrSxNoC&pg=PA300 - ↑ Leversha, Gerry; Smith, G. C. (November 2007), "Euler and triangle geometry",
*The Mathematical Gazette***91**(522): 436–452, doi:10.1017/S0025557200182087 - ↑ "An essay on the properties of triangles inscribed in and circumscribed about two given circles",
*Miscellanea Curiosa Mathematica***4**: 117–124, 1746, https://archive.org/details/miscellaneacuri01unkngoog/page/n142. 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,**36**, Mathematical Association of America, p. 56, ISBN 9780883853429, https://books.google.com/books?id=U1ovBsSRNscC&pg=PA56 - ↑
^{6.0}^{6.1}Svrtan, Dragutin; Veljan, Darko (2012), "Non-Euclidean versions of some classical triangle inequalities",*Forum Geometricorum***12**: 197–209, https://forumgeom.fau.edu/FG2012volume12/FG201217index.html; 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

## External links

- Weisstein, Eric W.. "Euler Triangle Formula". http://mathworld.wolfram.com/EulerTriangleFormula.html.

Original source: https://en.wikipedia.org/wiki/Euler's theorem in geometry.
Read more |