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Gergonne Point

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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.

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