Gibert Point


The Gibert point can be defined as follows. Given a reference triangle DeltaABC, reflect the point X_(1157) (which is the inverse point of the Kosnita point in the circumcircle) in each of the side lines of DeltaABC, to obtains the points A^', B^', and C^' to obtain the triangle DeltaA^'B^'C^'. The triangles DeltaABC and DeltaA^'B^'C^' are then perspective, with perspector given by the Gibert point, which is Kimberling center X_(1141).

The Gibert point has triangle center function


Consider the Neuberg cubic, which is the locus of a point M such that the reflections of M in the sidelines of a reference triangle DeltaABC are the vertices of a triangle perspective to DeltaABC. The locus of the perspector is the cubic K_n with trilinear equation


(Gibert). This cubic passes through Kimberling centers X_n for n=4, 5, 13, 14, 30, 79, 80, 265, 621, 622, 1117, and 1141.


The Gibert point is then also the unique point (other than A, B, and C) in which K_n meets the circumcircle.

The Gibert point lies on the line joining the nine-point center and Kosnita point (Grinberg 2003).

See also

Circumcircle, Triangle Cubic

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Ehrmann, J.-P.; and Gibert, B. "K060, K073: The K_n and K_i Cubics." §4.3.1 in "Special Isocubics in the Triangle Plane." Manuscript. pp. 66-67, July. 31, 2005., B. Hyacinthos posting #1498. 25 Sep 2000.Gibert, B. "K060: K_n=pK(X1989,X265)=O(X5)=D(infty).", D. "On the Kosnita Point and the Reflection Triangle." Forum Geom. 3, 105-111, 2003., C. "Encyclopedia of Triangle Centers: X(1141)=Gibert Point."

Referenced on Wolfram|Alpha

Gibert Point

Cite this as:

Weisstein, Eric W. "Gibert Point." From MathWorld--A Wolfram Web Resource.

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