Euler's theorem in geometry

$Euler's theorem: d = | I O | = R ( R − 2 r ) {\displaystyle d=|IO|={\sqrt {R(R-2r)}}}$

Euler's theorem:

d=|IO|={\sqrt {R(R-2r)}}

In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by^[1]^[2]

d^{2}=R(R-2r)

or equivalently

{\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},

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 1765.^[3] However, the same result was published earlier by William Chapple in 1746.^[4]

From the theorem follows the Euler inequality:^[5]^[6]

R\geq 2r,

which holds with equality only in the equilateral case.^[7]

Stronger version of the inequality

A stronger version^[7] is

{\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,

where

a

b

, and

c

are the side lengths of the triangle.

Euler's theorem for the escribed circle

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

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.^[8]

References

^ 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, 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, JSTOR 40378417
^ 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, 36, Mathematical Association of America, p. 56, ISBN 9780883853429
^ Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 124, ISBN 9781848165250
^ ^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