Lemoine Point triangle

Question

from Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How can I get the point (X,Y) of the lemoine point when I have only the vertexes of the triangle ? Thanks in advance.

Answer 1

Barycentric coordinates of the Symmedian point ($a^2:b^2:c^2$) together with conversion between barycentric and Cartesian coordinates provide a nice and simple solution for the Cartesian coordinates of the Symmedian point $K=(K_x,K_y)$ as a weighted average of the coordinates of vertices of $\triangle ABC,$ $A=(A_x,A_y),$ $B=(B_x,B_y)$ and $C=(C_x,C_y)$:

\begin{align} K=\frac{a^2\,A+b^2\,B+c^2\,C}{a^2+b^2+c^2}, \end{align}

where $a,b,c$ are lengths of corresponding sides of $\triangle ABC$, which can be easily obtained from known coordinates of the vertices.