‫מושגי יסוד במתמטיקה‬
‫ תשס"ח‬- '‫סמסטר ב‬
‫הגיאומטריה של הילברט‬
INCIDENCE AXIOMS
1 INCIDENCE AXIOM 1. For every point P and for every point Q not equal to P
there exists a unique line incident with
P
and
Q.
This line is denoted by
l(PQ).
2 INCIDENCE AXIOM 2. For every line
l there exist at least two distinct points
incident with l.
3 INCIDENCE AXIOM 3. There exist three distinct points with the property that no
line is incident with all three of them.
BETWEENNESS AXIOMS
A, B, and C are points, let the notation A*B*C
stand for the statement "point B is between point A and point C." If A and C
are points, the segment s(AC) is the set of points that includes A and C, and
all points B such that A*B*C. The points A and C are called the endpoints
of the segment s(AC).
4 BETWEENNESS AXIOM 1. If A*B*C, then A, B, and C are three distinct
points lying on the same line, and C*B*A.
5 BETWEENNESS AXIOM 2. Given any two distinct points B and D, there exist
points A, C, and E lying on l(BD) such that A*B*D, B*C*D, and
B*D*E.
6 BETWEENNESS AXIOM 3. If A, B, and C are three distinct points lying on the
NOTATION DEFINITION. If
same line, then one and only one of the points is between the other two.
l be any line, and A and B any points that do not lie on l. If A
equals B or if segment s(AB) contains no point lying on l, we say A and B are
on the same side of l, whereas if A is not equal to B and segment s(AB) does
intersect l, we say that A and B are on opposite sides of l.
7 SEPARATION AXIOM. For every line l and for any three points, A, B, and C
not lying on l:
DEFINITION. Let
‫הגיאומטריה של הילברט‬
‫מושגי יסוד במתמטיקה‬
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.
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.
CONGRUENCE AXIOMS
r(AB) is the following set of points lying on the line
l(AB): those points that belong to the segment s(AB) and all points C on
l(AB) such that B is between A and C. The ray r(AB) is said to emanate
from the vertex A and to be part of line l(AB). Rays r(AB) and r(AC)
from the same vertex A are opposite if they are distinct and if the lines l(AB)
and l(AC) are identical. An angle with vertex A is the point A together with
two distinct non-opposite rays with vertex A, say r(AB) and r(AC), that are
called the sides of the angle. Let a(BAC) denote this angle, with the
understanding that a(BAC) and a(CAB) are the same angle.
8 CONGRUENCE AXIOM 1. If A and B are distinct points and if A' is any point,
then for each ray r emanating from A' there is a unique point B' on r such that
B' is not equal to A' and s(AB) is congruent to s(A'B').
9 CONGRUENCE AXIOM 2. If s(AB) is congruent to s(CD) and s(AB) is
congruent to s(EF), the s(CD) is congruent to s(EF). Moreover, every
DEFINITIONS. The ray
segment is congruent to itself.
A*B*C, A'*B'*C', s(AB) is congruent
to s(A'B'), and s(BC) is congruent to s(B'C'), then s(AC) is congruent
to s(A'C').
11 CONGRUENCE AXIOM 4. Given any angle a(BAC) and given any ray
r(A'B') emanating from point A', there is a unique ray r(A'C') on a given
side of l(A'B') such that a(B'A'C') is congruent to a(BAC).
12 CONGRUENCE AXIOM 5. If angle a(...A...) is congruent both to angle
a(...B...) and to angle a(...C...), then a(...B...) is congruent to
a(...C...).
10 CONGRUENCE AXIOM 3. If
13 CONGRUENCE AXIOM 6 (Side-Angle-Side). 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.
 Leo Corry - 2011
-2-
‫סמסטר ב' תשס”ח‬
‫הגיאומטריה של הילברט‬
‫מושגי יסוד במתמטיקה‬
CONTINUITY AXIOMS
O and A. The set c(OA) of all points P such
that s(OP) is congruent to s(OA) is called a circle with O as center, and
each of the segments s(OP) is called a radius of the circle. If s(AB) and
s(CD) are two segments then s(AB) < s(CD) if there is a point E
between C and D such that s(AB) is congruent to s(CE). A point P is
inside (respectively, outside) a circle c(OA) if s(OP) < s(OA)
(respectively,s(OA) < s(OP)).
DEFINITIONS. Given two points
14 CONTINUITY AXIOM 1. If a circle has one point inside and one point outside
another circle, then the two circles intersect in two points.
s(CD) is any segment, A any point, and r any
ray with vertex A, then for every point B not equal to A on r there is a number
n such that when s(CD) is laid off n times on r starting at A, a point E is
reached such that ns(CD) is congruent to s(AE) and either B equals E or
B is between A and E.
15 CONTINUITY AXIOM 2. If
PARALLELISM AXIOM
16 PARALLELISM AXIOM. For every line l and every point P not lying on l there
is at most one line m through P such that m is parallel to l.
 Leo Corry - 2011
-3-
‫סמסטר ב' תשס”ח‬