证明一个四边形是矩形

Suppose we are given a quadrilateral $ABCD$. There are several easy ways to prove it is a rectangle.

1. Show that all angles are 90 degrees.
2. Show that one set of sides is parallel ,and that two opposite angles are 90 degrees.
3. Show that the diagonals bisect each other, and that they are equal in length.
4. Show that it is a parallelogram and that diagonals are equal in length.

These are quite redundant and they all go back to the original definition of a rectangle and its properties.