Saccheri quadrilateral

The summit angles of the Saccheri quadrilateral are 90 .

The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry.

The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure.

A Saccheri quadrilateral is a quadrilateral which has two sides of equal length, both perpendicular to a side called the base.

The Saccheri quadrilateral is now sometimes referred to as the Khayyam-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.

The close axiomatic study of Euclidean geometry led to the construction of the Lambert quadrilateral and the Saccheri quadrilateral.

He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral).

The first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, and it may occasionally be referred to as the Khayyam-Saccheri quadrilateral.

The Saccheri quadrilateral was first considered by Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.

So using a figure he found in Clavius, now called a Saccheri quadrilateral, Giordano tried to come up with his own proof of the assumption, in the course of which he proved:

A Lambert quadrilateral can be constructed from a Saccheri quadrilateral by joining the midpoints of the base and summit of the Saccheri quadrilateral.

Let ABCD be a Saccheri quadrilateral having AB as base, CA and DB the equal sides that are perpendicular to the base and CD the summit.

The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic.

Giordano Vitale, in his book Euclide restituo (1680, 1686), used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant.

Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid.