Is the point $P$ that minimizes $PA+PB+PC$ the orthocenter of $\Delta ABC$?

Question

I have a question about a triangle in ${\bf R}^2$. Let $\Delta ABC$ be a triangle with vertices $A(x_1,y_1)$, $B(x_2,y_2)$ and $C(x_3,y_3)$.

Question : Let $P$ be a point in the interior of $\Delta ABC $ which minimizes $PA + PB + PC$. Here $XY$ means the distance between $X$ and $Y$. What is $P$ ?

As you know, the center of mass minimizes $PA^2 + PB^2 + PC^2$. That is, the point is given by $(\frac{x_1+ x_2 + x_3 }{3}, \frac{y_1+ y_2 + y_3}{3})$

From this consideration we can think about such question. I guess that the point $P$ which minimizes $PA + PB + PC$ is an orthocenter. But I hav no evidence. Tell me anything about my conjecture or question. Thank you.

Answer 1

The point you're asking about is the Toricelli point, and it is in general distinct from the orthocenter. You can easily see that by taking an obtuse triangle. The orthocenter will be outside the triangle, so it certainly is not distance-minimizing.