Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.