证明一个四边形是菱形

When proving a quadrilateral is a rhombus, we must know several things:

1. A rhombus is a parallelogram with equal side lengths.

2. A rhombus has diagonals that are perpendicular to one another and bisect one another.

3. A rhombus' diagonals bisect the angles of the vertices each diagonal intersects.

Prove that when in a rectangle, the midpoints of the sides of the rectangle are drawn and labeled $A, B, C,$ and $D,$ then the quadrilateral $ABCD$ is a rhombus.

Say that the rectangle had side lengths of length $e$ and $f.$ Then the side lengths of quadrilateral $ABCD,$ by the Pythagorean Theorem, are $\sqrt{\left(\frac{e}{2}\right)^2+\left(\frac{f}{2}\right)^2}$.

Since all sides of this quadrilateral are equal then quadrilateral $ABCD$ must be a rhombus.