TOPICS

# Gibert Point

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

The Gibert point has triangle center function

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

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

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

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

Circumcircle, Triangle Cubic

## Explore with Wolfram|Alpha

More things to try:

## References

Ehrmann, J.-P.; and Gibert, B. "K060, K073: The and Cubics." §4.3.1 in "Special Isocubics in the Triangle Plane." Manuscript. pp. 66-67, July. 31, 2005. http://perso.wanadoo.fr/bernard.gibert/files/isocubics.html.Gibert, B. Hyacinthos posting #1498. 25 Sep 2000.Gibert, B. "K060: ." http://perso.wanadoo.fr/bernard.gibert/Exemples/k060.html.Grinberg, D. "On the Kosnita Point and the Reflection Triangle." Forum Geom. 3, 105-111, 2003. http://forumgeom.fau.edu/FG2003volume3/FG200311index.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(1141)=Gibert Point." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X1141.

Gibert Point

## Cite this as:

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