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Euler's Inequality

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The Euler triangle formula states that the distance d between the incenter and circumcenter of a triangle is given by

d^2=R(R-2r),

where R is the circumradius and r is the inradius. This immediately gives the inequality

R>=2r,

where equality holds iff the triangle is an equilateral triangle.

This inequality was published by Euler in 1765 (Bottema et al. 1969, p. 48). Mitrinovic et al. (1989) refer to it as the Chapple-Euler inequality.

See also

Circumradius, Euler Triangle Formula, Inradius

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References

Bottema, O.; Djordjevic, R. Z.; Janic, R.; Mitrinovic, D. S.; and Vasic, P. M. Geometric Inequalities. Groningen: Wolters-Noordhoff, p. 48, 1969.Kazarinoff, N. D. Geometric Inequalities. New York: Random House, pp. 78-84, 1961.Mitrinovic, D. S.; Pecaric, J. E.; and Volenec, V. Recent Advances in Geometric Inequalities. Dordrecht, Netherlands: Kluwer, 1989.

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Euler's Inequality

Cite this as:

Weisstein, Eric W. "Euler's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersInequality.html

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