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Gergonne Point

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GergonnePoint

The Gergonne point Ge is the perspector of a triangle DeltaABC and its contact triangle DeltaT_AT_BT_C. It has equivalent triangle center functions

alpha=[a(b+c-a)]^(-1)
(1)
alpha=sec^2(1/2A)
(2)

and is Kimberling center X_7.

GergonneNagelConjugates

The Gergonne point Ge is the isotomic conjugate of the Nagel point Na. The Gergonne point of a triangle is the symmedian point of its contact triangle (Honsberger 1995).

MittenpunktCollinear

The Gergonne point Ge, triangle centroid G, and mittenpunkt M are collinear, with GeG:GM=2:1.

Distances from some other named triangle centers include

GeI=(4ILr^2)/(a^2-2ab+b^2-2ac-2bc+c^2)
(3)
GeL=(2(a^3-ba^2-ca^2-b^2a-c^2a-2bca+b^3+c^3-bc^2-b^2c)IL)/((a+b+c)(a^2-2ba-2ca+b^2+c^2-2bc))
(4)
GeNa=(4(a^2+b^2+c^2)IK)/(a^2-2ab+b^2-2ac-2bc+c^2),
(5)

where I is the incenter, L is the de Longchamps point, and Na is the Nagel point.

See also

Adams' Circle, Contact Triangle, Gergonne Line, Nagel Point

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References

Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 160-164, 1952.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. New York: Random House, pp. 11-13, 1967.Eves, H. W. A Survey of Geometry, rev. ed. Boston, MA: Allyn and Bacon, p. 83, 1972.Gallatly, W. "The Gergonne Point." §32 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 22, 1913.Honsberger, R. "The Gergonne Point." §7.4 (iv) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 61-62, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 184 and 216, 1929.Kimberling, C. "Gergonne Point." http://faculty.evansville.edu/ck6/tcenters/class/gergonne.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(7)=Gergonne Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X7.

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Gergonne Point

Cite this as:

Weisstein, Eric W. "Gergonne Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GergonnePoint.html

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