Euler's theorem in geometry
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)
- Egan conjecture, generalization to higher dimensions
- 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.
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