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
Bicentric quadrilateral the circumcircle is the vertex of a bicentric quadrilateral having the same incircle and circumcircle. This is a corollary of Poncelet's porism, which was proved by the French mathematician Jean-Victor Poncelet (1788–1867). Examples of bicentric quadrilaterals are squares, right kites, and isosceles tangential trapezoids. A convex quadrilateral "ABCD" with sides "a", "b", "c", "d" is bicentric if and only if opposite sides satisfy Pitot's theorem for tangential quadrilaterals "and" the cyclic quadrilateral property that opposite angles are supplementary; that is, Three other characterizations concern the points where the incircle in a tangential quadrilateral is tangent to the sides. If the