Euler's theorem in geometry

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[1][2]

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

From the theorem follows the Euler inequality:[5]

R ≥ 2 r , {\displaystyle R\geq 2r,}
which holds with equality only in the equilateral case.[6]

Stronger version of the inequality

A stronger version[6] 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.[7]

See also

References

  1. ^ Johnson, Roger A. (2007) [1929], 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. ^ a b 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

External links

Wikimedia Commons has media related to Euler's theorem in geometry.
  • Weisstein, Eric W., "Euler Triangle Formula", MathWorld