How to prove the property of the Lemoine point of a triangle? 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. !Lemoine point How to prove such a property?

Let $(d_a,d_b,d_c)$ be the distances of your point from the sides. You have to minimize the quantity: $$ d_a^2+d_b^2+d_c^2 $$ under the constraint: $$ a d_a + b d_b + c d_c = 2\Delta,$$ hence Lagrange multipliers gives that $(d_a,d_b,d_c)=\lambda(a,b,c)$, so your stationary point is the isogonal conjugate of the point having trilinear coordinates $\left(\frac{1}{a},\frac{1}{b},\frac{1}{c}\right)$, that is the centroid.