1. ## minimizing squares...

Given a triangle $\displaystyle \triangle ABC$ and a straight line $\displaystyle l$, find the point $\displaystyle P$ on $\displaystyle l$ such that $\displaystyle (PA)^2 + (PB)^2 + (PC)^2$ is the smallest.

2. Hello,
Originally Posted by scipa
Given a triangle $\displaystyle \triangle ABC$ and a straight line $\displaystyle l$, find the point $\displaystyle P$ on $\displaystyle l$ such that $\displaystyle (PA)^2 + (PB)^2 + (PC)^2$ is the smallest.
Let $\displaystyle A_h,\,B_h$ and $\displaystyle C_h$ be the orthogonal projection of $\displaystyle A,\,B$ and $\displaystyle C$ on $\displaystyle l$, respectively. Using Pythagorean theorem :

$\displaystyle \begin{cases} PA^2=PA_h^2+AA_h^2\\ PB^2=PB_h^2+BB_h^2\\ PC^2=PC_h^2+CC_h^2\\ \end{cases}$

hence

$\displaystyle PA^2+PB^2+PC^2=\underbrace{AA_h^2+BB_h^2+CC_h^2}_{ \text{that's a constant}}+PA_h^2+PB_h^2+PC_h^2$

As $\displaystyle AA_h^2+BB_h^2+CC_h^2$ doesn't depend on $\displaystyle P$ and $\displaystyle PA_h^2+PB_h^2+PC_h^2\geq 0$, minimizing $\displaystyle PA^2+PB^2+PC^2$ is equivalent to minimizing $\displaystyle PA_h^2+PB_h^2+PC_h^2$. How can this be achieved ?