Neuberg cubic (K001)

We construct triangle

and then K001 cubic. After that we take point

which can be any point from the cubic then we cojugate that point isogonal and we get point

which is also on the same cubic. And by using GeoGebra we conclude that K001 is isogonal transformed of itself.

Barycentric equation

Let the barycentric coordinates of a point are

and

. The barycentric coordinates of its isogonal conjugate are

and

. And now we substitude

and

in the equation with

and

In this we get the same equqtion as that in the beginig and this means that Neuberg cubic is isogonal transform of itself.