Proving a Quadrilateral Is a Square

We all know that a square is a quadrilateral where all four sides are equal in length and all four angles are the same; namely $90^\circ$. To prove that a particular quadrilateral is a square, verify both these conditions. This is the easiest when we have coordinates. Simply use the slopes of each segment to verify the right angles. Then, use the distance formula to compare the lengths of the segments.