# Math Help - Minimisation problem

1. ## Minimisation problem

I got stuck on this question and I hope somebody could help

Given the triangle ABC and a point P lies inside the triangle, show that the distance AP^2 + BP^2 + CP^2 is minimum. Show further that P is the intersection point of three madians of the triangle.

Thank you for your time

KN

2. Originally Posted by knguyen2005
Given the triangle ABC and a point P lies inside the triangle, show that the distance AP^2 + BP^2 + CP^2 is minimum. Show further that P is the intersection point of three madians of the triangle.
I don't see an elegant geometric way to do this. Analytically, it's easy but ugly.

Let A, B, C be the points $(a_1,a_2),\ (b_1,b_2),\ (c_1,c_2)$, and let $P=(x,y)$. Then $PA^2 = (x-a_1)^2 + (y-a_2)^2$, with similar expressions for $PB^2$ and $PC^2$. Add these together and you get $3x^2 - 2(a_1+b_1+c_1)x + 3y^2 - 2(a_2+b_2+c_2)y + \text{const.}$ This is equal to $3\bigl(x - \tfrac13(a_1+b_1+c_1)\bigr)^2 + 3\bigl(y - \tfrac13(a_2+b_2+c_2)\bigr)^2 + \text{const.}$, which is minimised when $x = \tfrac13(a_1+b_1+c_1)$ and $y = \tfrac13(a_2+b_2+c_2)$.