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

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.

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I took the liberty of editing your title to be more specific. I hope you don't mind. –  Rahul Sep 2 '12 at 1:31