Quadrilateral

A Euclidean polygon with four sides and four right angles. Opposite sides are parallel and have equal lengths. A rectangle is a square exactly when its adjacent side lengths agree.

A rectangle with all four side lengths equal. It is a regular four-sided polygon. Rotations and translations preserve this definition; its sides need not be parallel to the coordinate axes.

A parallelogram is the convex set with vertices $p,p+u,p+v,p+u+v$ for two linearly independent planar vectors $u,v$. Its opposite edges are parallel and its centre $p+(u+v)/2$ gives central symmetry. Its area is $|\det [u,v]|$, so an invertible linear map changes its area by the absolute value of that map's determinant. These facts make parallelograms convenient convex bodies for two-dimensional applications of the Minkowski convex body theorem.

For a quadrilateral with side lengths $a,b,c,d$, semiperimeter $s$, area $K$, and opposite angles $A,C$,The second term vanishes exactly when $A+C=\pi$, which is the cyclicity condition for a nondegenerate convex quadrilateral.