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 by 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.