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geometry > Euclidean geometry > Euler's theorem in geometry

Preferred term

Euler's theorem in geometry

Definition(s)

In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by
${\\displaystyle d^{2}=R(R-2r)}$
or equivalently
${\\displaystyle {\ rac {1}{R-d}}+{\ rac {1}{R+d}}={\ rac {1}{r}},}$
where ${\\displaystyle R}$ and ${\\displaystyle r}$ 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:
${\\displaystyle R\\geq 2r,}$
which holds with equality only in the equilateral case.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Euler%27s_theorem_in_geometry)

Broader concept(s)

Euclidean geometry

Related concept(s)

triangle

In other languages

relation d'Euler dans le triangle

French
relations d'Euler dans le triangle

URI

http://data.loterre.fr/ark:/67375/PSR-JP1PHSKD-9

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