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Euler's theorem in geometry

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On distance between centers of a triangle
Euler's theorem:
d = | I O | = R ( R − 2 r ) {\displaystyle 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 d 2 = R ( R − 2 r ) {\displaystyle d^{2}=R(R-2r)} or equivalently 1 R − d + 1 R + d = 1 r , {\displaystyle {\frac {1}{R-d}}+{\frac {1}{R+d}}={\frac {1}{r}},} where R {\displaystyle R} and r {\displaystyle 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. However, the same result was published earlier by William Chapple in 1746.

From the theorem follows the Euler inequality: R ≥ 2 r , {\displaystyle R\geq 2r,} which holds with equality only in the equilateral case.

Stronger version of the inequality

A stronger version 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 escribed 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.

See also

References

  1. Johnson, Roger A. (2007) , Advanced Euclidean Geometry, Dover Publ., p. 186
  2. 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
  3. 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
  4. 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.
  5. 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
  6. ^ Svrtan, Dragutin; Veljan, Darko (2012), "Non-Euclidean versions of some classical triangle inequalities", Forum Geometricorum, 12: 197–209; see p. 198
  7. 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

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