Lucas cubic (K007)

Here we construct a triangle

. After that we construct Lucas cubic for this triangle and we take point

on it then we conjugate

isogonal and we get point

. By using GeoGebra we get another cubic, the blue one one the figure. Lucas cubic is isogonal transfom of K172.

Barycentric equation

Here we again sustitude

and

with

and

But the barycentric equation of the isogonal transform cubic is

where

and in order to get the same equation we have to multiply

and the result is-

and this is the same equation as that of the isogonal transform cubic which means that Lucas cubic is isogonal transform of K172.