Hilbert's Axioms are divided into
sections: Incidence Axioms, Betweenness
Axioms, Congruence Axioms, Continuity Axiom,
and Parallelism Axiom.
Incidence Axioms:
HI1. For every point P and for
every point Q not equal to P, there exists
a unique line
incident with P and Q.
HI2. For every line
there exists at least two distinct points
that are incident with .
HI3. There exist three distinct points
with the property that no line is incident
with all three of them.
Betweenness Axioms  the symbol "*"
means "lies between"
HB1. If A * B * C, then A, B, and
C are three distinct points all lying on
the same line,
and C * B * A.
HB3. If A, B, and C are three distinct
points lying on the same line, then one
and only one of the points is between the
other two.
HB4. For every line
and for any three points A, B, and C
not lying on :
(i) If A and B are on the same
side of
and
B and C are on the same side of
,
then A and C are on the same side of
.
(ii) If A and B are on opposite
sides of
and
B and C are on opposite sides of
,
then A and C are on the same side of
.
Congruence Axioms
HC5. If < A is congruent to < B
and < A is congruent to < C, then
< B is congruent to < C.
Moreover, every angle is congruent to
itself.
HC6. If two sides and the included
angle of one triangle are congruent
respectively to two
sides and the included angle of another
triangle, then the two triangles are
congruent.
Continuity Axiom
Parallelism Axiom
For every line
and every point P not lying on
there is at most one line m through
P such that m is parallel to
.
Euclidean and
NonEuclidean Geometries: Development and
History
by M. J. Greenberg
pp. 469  471
