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 argument with the other two points of tangency completes the proof of the result. If the extensions of opposite sides in a tangential quadrilateral intersect at "J" and "K", and the diagonals intersect at "P", then "JK" is perpendicular to the extension of "IP" where "I" is the incenter. The incenter of a tangential quadrilateral lies on its Newton line (which connects the midpoints of the diagonals). The ratio of two opposite sides in a tangential quadrilateral can be expressed in terms of the distances between the incenter "I" and the vertices according to The product of two adjacent sides