Rhombus

Definition

Simple Definition

A quadrilateral with all sides of equal length is called a rhombus.

Linear Algebraic Definition

For two distinct vectors $\mathbf{a}$ and $\mathbf{b}$ of equal magnitude on a 2D Cartesian plane, the following set is called a rhombus.

$$ R = \left\{ \lambda_{1}\mathbf{a} + \lambda_{2} \mathbf{b} : 0\leq \lambda_{1},\lambda_{2}\leq 1, \quad |\mathbf{a}| = |\mathbf{b}|, \quad \mathbf{a} \ne \mathbf{b} \right\} $$

Explanation

By definition, if it is a square, then it is a rhombus. If it is a rhombus, then it is a parallelogram and a trapezoid.

Square

$\implies$ Rectangle $\implies$

$\implies$ Rhombus $\implies$

Parallelogram $\implies$ Trapezoid

If the condition of having equal angles (rectangle) is added, it becomes a square.