In geometry, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral),^{[1]}^{[2]} is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists. In hyperbolic geometry the fourth angle is acute, in Euclidean geometry it is a right angle and in elliptic geometry it is an obtuse angle.

A Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral. This line segment is perpendicular to both the base and summit and so either half of the Saccheri quadrilateral is a Lambert quadrilateral.

Lambert quadrilateral in hyperbolic geometry

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In hyperbolic geometry a Lambert quadrilateral AOBF where the angles$\angle FAO,\angle AOB,\angle OBF$ are right, and F is opposite O ,$\angle AFB$ is an acute angle, and the curvature = -1 the following relations hold:^{[3]}

^Rashed, Roshdi; Papadopoulos, Athanase (2017-10-23). Menelaus' 'Spherics': Early Translation and al-Māhānī / al-Harawī's Version. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-056987-2.

^the alternate name Ibn al-Haytham–Lambert quadrilateral, has been suggested in Boris Abramovich Rozenfelʹd (1988), A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, p. 65. Springer, ISBN 0-387-96458-4, in honor of Ibn al-Haytham

^Martin, George E. (1998). The foundations of geometry and the non-Euclidean plane (Corrected 4. print. ed.). New York, NY: Springer. p. 436. ISBN 0387906940.

References

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George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975

M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History, 4th edition, W. H. Freeman, 2008.