In geometry, Euler's theorem relates the distance d between the circumcenter and incenter of a triangle by the formula d² = R(R - 2r), where R and r are the circumradius and inradius, the radii of the circumscribed circle and inscribed circle, respectively. Equivalently, it satisfies 1/(R - d) + 1/(R + d) = 1/r. Named after Leonhard Euler, who published it in 1765, the result was also found earlier by William Chapple in 1746. This theorem leads to the Euler inequality R ≥ 2r, which holds with equality only in the equilateral triangle case.
Stronger version of the inequality
A stronger version7 is R r ≥ a b c + a 3 + b 3 + c 3 2 a b c ≥ a b + b c + c a − 1 ≥ 2 3 ( a b + b c + c a ) ≥ 2 , {\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,} where a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are the side lengths of the triangle.
Euler's theorem for the excribed circle
If r a {\displaystyle r_{a}} and d a {\displaystyle d_{a}} denote respectively the radius of the escribed circle opposite to the vertex A {\displaystyle A} and the distance between its center and the center of the circumscribed circle, then d a 2 = R ( R + 2 r a ) {\displaystyle 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
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)
- Egan conjecture, generalization to higher dimensions
- List of triangle inequalities
External links
Wikimedia Commons has media related to Euler's theorem in geometry.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, vol. 2, Mathematical Association of America, p. 300, ISBN 9780883855584 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 /wiki/The_Mathematical_Gazette ↩
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. /wiki/William_Chapple_(surveyor) ↩
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 9780883853429 ↩
Svrtan, Dragutin; Veljan, Darko (2012), "Non-Euclidean versions of some classical triangle inequalities", Forum Geometricorum, 12: 197–209; see p. 198 https://forumgeom.fau.edu/FG2012volume12/FG201217index.html ↩
Svrtan, Dragutin; Veljan, Darko (2012), "Non-Euclidean versions of some classical triangle inequalities", Forum Geometricorum, 12: 197–209; see p. 198 https://forumgeom.fau.edu/FG2012volume12/FG201217index.html ↩
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 /wiki/Doi_(identifier) ↩