Parallelogram Parallelogram In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The
Tangential quadrilateral sides "a, b, c, d" and the successive tangent lengths "e, f, g, h" as Since "eg" = "fh" if and only if the tangential quadrilateral is also cyclic and hence bicentric, this shows that the maximal area formula_10 occurs if and only if the tangential quadrilateral is bicentric. A trigonometric formula for the area in terms of the sides "a", "b", "c", "d" and two opposite angles is For given side lengths, the area is maximum when the quadrilateral is also cyclic and hence a bicentric quadrilateral. Then formula_12 since opposite angles are supplementary angles. This can be proved