Find the point on the plane xOy

Question

Let $A(x_1; y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$ be three points not lying on the same straight line. Find the point on the plane $xOy$ such that the sum of the distances from it to these points is the least.

I got the answer, but as for me, it is incorrect because the optimization of another function.

Answer 1

We want to minimise

$$D(x,y)=\sum_{i =1}^3(x-x_i)^2+(y-y_i)^2$$

This happens when

$$\begin{align} {\partial{D}\over\partial{x}}&=0\\{\partial{D}\over\partial{y}}&=0\end{align}$$

And this means

$$\begin{align} x&={x_1+x_2+x_3\over 3}\\y&={y_1+y_2+y_3\over 3}\end{align}$$