Plane Geometry - Bruce E. Shapiro

Section 31Quadrilaterals in NeutralGeometryWe will see in a later section that there are no rectangles in neutralgeometry. In fact, the existence of a rectangle is equivalent to assumingthe Euclidean parallel postulate (theorem 33.3).Starting as early as the 11th century parallelograms with two right angles,and quadrilaterals with three right angles, were studied in many attemptsto prove that the remaining interior angles were right angles. Much of thework, which goes back to Umar al-Khayyami (1048-1131) and Nasir Eddin(1201-1274) is often attributed to the European geometers GiovanniSaccheri (1667-1733) and Johann Lambert (1728-1777) who rediscoveredthe earlier results. None of these attempts were able to successfully demonstratethat a rectangle could be constructed using only the axioms of neutralgeometry.Here we look at some of the results that can be obtained for quadrilateralsin Neutral Geometry.Definition 31.1 Let A, B, C, D be points, no 3 of which are collinear, suchthat any two of the segments AB, BC, CD, DA either have no point incommon or only have an endpoint in common. Then the points A, B, C, Ddetermine a quadrilateral, denoted by □ABCD.The points A, B, C, D are called the vertices of the quadrilateral.The segments AB, BC, CD, DA are called the sides of the quadrilateral.The diagonals of □ABCD are the segments AC and BD.145