We can prove this as follows :
Connect all four midpoints
Distance from (-a, b) to (a,b) = 2a
Distance from (-a, -b) to (a, - b) = 2a ....so these opposite sides are congruent
Distance from (-a, b) to (-a, -b) = 2b
Distance from (a,b) to ( a, -b) = 2b......so these opposite sides are congruent
And the slope between (-a,b) and (a,b) = [b - b] / [-a - a] = 0
And the slope between (-a,-b) and (a, -b) = [ - b - -b] / [ -a -a] = 0.....so.....these opposite sides are parallel
And the slope between (-a,b) and (-a, -b) = [-b - b] / [ -a - - a] = is undefined
And the slope between (a,b) and (a,-b) = [-b - b] / [a - a] is undefined....so these opposite sides are parallel
And the diagonals drawn from (-a,b) to (a, -b) and from (-a,-b) to (a,b) meet at the origin and they bisect each other since the distance from the origin to each of these points = sqrt(a^2 + b^2)
So...the opposite sides are congruent and parallel and the diagonals bisect each other.....thus.....this is a rectangle
