SE-SKOLORNAS 2010 Problem 2

Consider the four lines $y = mx-k^2$ for different integer $k$ . Let $(x_i,y_i)$ , $i = 1,2,3,4$ be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat $x_1\leq x_2\leq x_3\leq x_4$ . Show that $x_1 + x_4 =x_2+x_3$ and $y_1y_4 =y_2y_3$ .