From: wilson@web.ctron.com (David Wilson)
Newsgroups: sci.math
Subject: Re: Parallel axiom.
Date: 28 Aug 92 13:39:49 GMT
Reply-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."