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Euler Triangle Formula

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The Euler triangle formula relates the distance between a triangle's circumcenter and incenter to its circumradius and inradius. Let O and I be the circumcenter and incenter of a triangle with circumradius R and inradius r. Let d be the distance between O and I. Then

d^2=R(R-2r)

(Mackay 1886-1887; Casey 1888, pp. 74-75; Johnson 1929, pp. 186-187; Altshiller-Court 1952, p. 85). This is the simplest case of Poncelet's porism, and is sometimes also known as Euler's triangle theorem (Altshiller-Court 1952, p. 85).

From this theorem, the inequality

R>=2r,

sometimes known as Euler's inequality, follows immediately.

See also

Circumcenter, Circumradius, Euler's Inequality, Incenter, Inradius, Poncelet's Porism

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References

Altshiller-Court, N. "Euler's Theorem." §152 in College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 85-86, 1952.Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 6th ed. Dublin: Hodges, Figgis, & Co., 1888.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kazarinoff, N. D. Geometric Inequalities. New York: Random House, pp. 78-79, 1961.Mackay, J. S. "Historical Notes on a Geometrical Theorem and Its Developments [18th Century]." Proc. Edinburgh Math. Soc. 5, 62-78, 1886-1887.

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Euler Triangle Formula

Cite this as:

Weisstein, Eric W. "Euler Triangle Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulerTriangleFormula.html

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