From:wilson@web.ctron.com (David Wilson)Newsgroups:sci.mathSubject:Re: Parallel axiom.Date:28 Aug 92 13:39:49 GMTReply-To:wilson@web.ctron.com (David Wilson)Organization:Cabletron Systems INc.

The book "The Foundations of Geometry and the Non-Euclidean Plane" by George E. Martin lists the following 26 equivalents to the Parallel Postulate within absolute geometry: Proposition A. Euclid's Parallel Postulate: If A and D are points on the same side of segment(BC) such that measure(angle(ABC)) + measure(angle(BCD)) < pi, then ray(BA) intersects ray(CD). Proposition B. Euclid's Proposition I.29: If A and D are points on the same side of line(BC) and line(BA) || line(CD), then measure(angle(ABC)) + measure(angle(BCD)) = pi. Proposition C. Euclid's Proposition I.30: l || m and m || n implies l || n for lines l, m, n. (Lines parallel to a given line are parallel.) Proposition D. Contrapositive to Euclid's Proposition I.30: A third line intersecting one of two parallel lines intersects the other. Proposition E. Euclid's Proposition I.31, Playfair's Parallel Postulate: If a point P is off line l, then there exists a unique line through P parallel to l. Proposition F. A line perpendicular to one of two parallel lines is perpendicular to the other. Proposition G. l || m, r is perpendicular to l, and s is perpendicular to m implies r || s for lines l, m, r, s. Proposition H. The perpendicular bisectors of the sides of a triangle are concurrent. Proposition I. There exists a circle passing through any three noncollinear points. Proposition J. There exists a point equidistant from any three noncollinear points. Proposition K. A line intersecting and perpendicular to one ray of an acute angle intersects the other ray. Proposition L. Through any point in the interior of an angle there exists a line intersecting both rays of the angle not at the vertex. Proposition M. Euclid's Proposition I.32: The sum of the measures of the angles of any triangle is pi. The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Proposition N. Theorem of Thales: If point C is off segment(AB), but on the circle with diameter segment(AB), then angle(ABC) is right. Proposition O. If angle(ABC) is right, then C is on the circle with diameter segment(AB)> Proposition P. The perpendicular bisectors of the legs of a right triangle intersect. Proposition Q. l is perpendicular to r, r is perpendicular to s, and s is perpendicular to m implies l intersects m for all l, m, r, s. Proposition R. There exists an acute angle such that every line intersecting and perpendicular to one ray of the angle intersects the other ray. Proposition S. There exists an acute angle such that every point in the interior of the angle is on a line intersecting both rays of the angle. Proposition T. There exists one triangle such that the sum of the measures of its angles is pi. Proposition U. There exists one triangle with defect 0. [The defect of a triangle is pi less the sum of the measures of the angles of the triangle. This is a restatement of Proposition T]. Propositoin V. Saccheri's Hypothesis of the Right Angle: There exists a rectangle. [The Hypothesis of the Acute (Obtuse) Angle: There exists a quarilateral with two adjacent right angles and two adjacent acute (obtuse) angles. These apply to the hyperbolic and elliptic planes, respectively.] Proposition W. There exist two lines l and m such that l is equidistant from m [all points of l are at a constant positive distance from m]. Proposition X. If three angles of a quadrilateral are right, then so is the fourth. Proposition Y. There is some line l and some point P off l such that a unique line parallel to l passes through P. Proposition Z. There exist a pair of similar noncongruent triangles. -- David W. Wilson (wilson@ctron.com) Disclaimer: "Truth is just truth...You can't have opinions about truth." - Peter Schikele, introduction to P.D.Q. Bach's oratorio "The Seasonings."