Ihad the question: "four points a, b, c and d lie on the curve with equation y = x^2. prove that abcd is never a parallelogram." i have a proof but i don't really think it's strong enough: co-ordinates for points of the parallelogram from left to right: (x3, y2), (x1, y1), (x4, y2), (x2, y1) y1 = x1^2 y1 = x2^2 y2 = x3^2 y2 = x4^2 x1 > x3 x2 > x4 x1^2 = x2^2 x1 ± x2 x3^2 = x4^2 x3 ± x4 as they are all different points: x1 = -x2 and x3 = -x4 or -x1 = x2 and -x3 = x4 so: x1 > x3 and -x1 > -x3 but x1 > x3 is equivalent to -x1 < -x3 -x1 > -x3 and -x1 < -x3 can't both be true therefore:   abcd will never be a parallelogram is that valid? or could i make it stronger?