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
![sum_(cyclic)aalphaS_A[c^2gamma^2(4S_C^2-a^2b^2)-b^2beta^2(4S_B^2-c^2a^2)]](/images/equations/GibertPoint/NumberedEquation2.svg) |
(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).
and 
Cubics." §4.3.1 in "Special Isocubics in
the Triangle Plane." Manuscript. pp. 66-67, July. 31, 2005. 
."