Heres my scenario.. I have two parallel lines segments,AB and CD of equal length. If A,B are known, how do I find C and D?!
$x_3,y_3$ can be arbitrary. Then let $x_4=x_3+x_2-x_1, y_4=y_3+y_2-y_1$.
If you more specifically want $ABDC$ to be a rectangle, select $t$ arbitrary, then let $x_3=x_1+t(y_2-y_1), y_3=y_1+t(x_1-x_2)$ in the above.