AXIOMS

INCIDENCE AXIOMS:

I-1

For every point P and for every point Q not equal to P, there is a unique line l incident with P and Q.

I-2

For every line l there exist at least two distinct points that are incident with l.

I-3

There exist three distinct points with the property that no line is incident with all three of them.

 

BETWEENNESS AXIOMS:

B-1

If A*B*C, then A, B, and C are three distinct points all lying on the same line, and C*B*A.

B-2

Given any two distinct points B and D, there exist points A, C, and E lying on such that A*B*D, B*C*D, and B*D*E.

B-3

If A, B, and C are three distinct points lying on the same line, then one and only one of these points is between the other two.

B-4

(Plane Separation Axiom) For any line l, and for any three points A, B, and C not lying on l:

(i) If A and B are on the same side of l and B and C are on the same side of l, then A and C are on the same side of l.

(ii) If A and B are on opposite sides of l and B and C are on opposite sides of l, then A and C are on the same side of l.

(iii)* If A and B are on opposite sides of l and B and C are on the same side of l, then A and C are on opposite sides of l.

* (iii) is not an axiom, but a corollary of the Plane Separation Axiom (i) and (ii). However, due to the similarity between the three statements, I decided to state (iii) here, rather than with the other propositions.

 

 CONGRUENCE AXIOMS:

C-1

If A and B are distinct points and if C is any point, then for each ray r emanating from C there is a unique point D on r such that C and D are distinct and .

C-2

If and then . Moreover each segment is congruent to itself.

C-3

(Segment Addition) If A*B*C, D*E*F, , and , then .

C-4

Given any angle , and any given ray emanating from a point D, then there is a unique ray on a given side of line such that angles and are congruent.

C-5

If and , then . Moreover every angle is congruent to itself.

C-6

(SAS) If two sides and the included angle of a triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.

 

  

 CONTINUITY AXIOMS:

Circular Continuity Principle*

If a circle has a point inside and a point outside another circle, then the two circles intersect in two points.

Elementary Continuity Principle*

If one endpoint of a segment is inside a circle and the other outside, then the segment intersects the circle.

Archimedes' Axiom*

If CD is any segment, A any point, and r any ray with vertex A, then for any point B on r different from A, there is a positive integer n, such that when CD is laid off n times on r starting at A, a point E is reached such that and either B=E or A*B*E.

Aristotle's Axiom*

Given any side of an acute angle and any segment AB, there exists a point Y on the given side of the angle such that if X is the foot of the perpendicular from Y to the other side of the angle, XY>AB.

Corollary*

Given any acute angle , and any given ray , and a point P not collinear with D and E, there is a point R on ray such that .

Dedekind's Axiom

Suppose that the set of points of a line l is a disjoint union of two nonempty subsets S and T, such that no points of either subset is between two points of the other. Then there is a unique point O on l such that one of the subsets is equal to a ray of l with vertex O, and the other subset is equal to the complement.

* Although Dedekind's axiom implies the other continuity principles and is the only continuity axiom we need to assume, we still refer to the others as "axioms".

PARALLELISM AXIOMS:

Hilbert's Parallel Axiom*

For every line l and point P not lying on l there is at most one line m through P such that m is parallel to l.

Euclid Fifth Postulate*

If two lines are intersected by a transversal in such a way that the sum of the degree measures of the two interior angles on one side of the transversal is less than 1800, then the two lines meet on that side of the transversal.

Hyperbolic Parallel Axiom**

There exist a line l and a point P not on l such that at least two distinct lines parallel to l pass through P.

*Hilbert's Parallel Axiom and Euclid's Fifth Postulate are equivalent. Any one of these axioms together with the axioms of Neutral geometry (incidence axioms I-1, I-2, I-3; betweenness axioms B-1, B-2, B-3, B-4; congruence axioms C-1, C-2, C-3, C-4, C-5, C-6; and Dedekind's continuity axiom) give a complete system of axioms for the Euclidean geometry.

**Hyperbolic Parallel Axiom (the negation of Hilbert's Parallel Axiom) together with the axioms of Neutral geometry (incidence axioms I-1, I-2, I-3; betweenness axioms B-1, B-2, B-3, B-4; congruence axioms C-1, C-2, C-3, C-4, C-5, C-6; and Dedekind's continuity axiom) give a complete system of axioms for the Hyperbolic geometry.

 

SEPARATION AXIOMS:*

S-1

If (A, B|C, D), then points A, B, C and D are collinear and distinct.

S-2

If (A, B|C, D), then (C, D|A, B) and (B, A|C, D).

S-3

If (A, B|C, D), then not (A, C|B, D).

S-4

If points A, B, C, and D are collinear and distinct, then:

(A, B|C, D) or (A, C|B, D) or (A, D|B, C).

S-5

If points A, B, and C are collinear and distinct, then there exists a point D, such that (A, B|C, D).

S-6

For any five distinct collinear points A, B, C, D, and E, if (A, B|D, E) then either (A, B|C, D) or (A, B|C, E).

S-7

Perspectivities preserve separation; i.e., if (A, B|C, D), with l the line through A, B, C, and D, and P, Q, R, and S are the corresponding points on line m under a perspectivity, then (P, Q|R, S).

* in elliptic geometry the betweenness axioms are replaced with separation axioms.