Thread: 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

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 , and let . Then , with similar expressions for and . Add these together and you get This is equal to , which is minimised when and .