Square

Definition

Simple Definition

A quadrilateral with all sides of equal length and all angles of equal measure is called a square.

Linear Algebraic Definition

On a 2-dimensional coordinate plane, for two perpendicular and equal magnitude vectors $\mathbf{a}$ and $\mathbf{b}$, the following set is called a square.

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

Explanation

By definition, a square is a rectangle, a parallelogram, a rhombus, and a trapezoid.

Square

$\implies$ Rectangle $\implies$

$\implies$ Rhombus $\implies$

Parallelogram $\implies$ Trapezoid

Without the condition that all angles are equal, it is a rhombus, and without the condition that all sides are of equal length, it is a rectangle.