Originally Posted by

**Laurent** Hi,

I'd rather suggest you procede like this: $\displaystyle P_1P^2-P_2P^2=(\overrightarrow{P_1P}-\overrightarrow{P_2P})\cdot (\overrightarrow{P_1 P}+\overrightarrow{P_2 P})$ (like $\displaystyle a^2-b^2=(a-b)(a+b)$ for dot-product). The first factor is $\displaystyle \overrightarrow{A}$, and you can leave the second factor as such (or replace it by $\displaystyle 2\overrightarrow{IP}$ where $\displaystyle I$ is the middle of $\displaystyle [P_1,P_2]$). Do the same for $\displaystyle P_3,P_4$, and sum both results to make $\displaystyle \overrightarrow{B}$ appear.

The problem with your method is that you're using $\displaystyle \|P_1\|^2$, which probably means $\displaystyle \|\overrightarrow{OP_1}\|^2$ for some point $\displaystyle O$? Thus you are introducing a new point, while you should be looking for relation between the four corner points only. In the situation of the problem, you should forbid yourself to write things like $\displaystyle \|P_1\|^2$, which only make sense with respect to a given origin.