MAT 360 Lecture 5
Hilbert’s axioms - Betweenness
EXERCISE:
 Can
you deduce from the Incidence
Axioms that there exist one point
and one line?
 Can you deduce from the Euclid’s I
to V Axioms that there exist one
point and one line?
2
Incidence Axioms
1. For each point P and for each point Q not equal
to P there exists a unique line incident with P and
Q.
2. For every line T there exist at least two distinct
points incident with T.
3. There exist three distinct points with the property
that no line is incident with all the three of them.
Euclid’s postulates
I.
For every point P and every point Q not equal to P
there exists a unique line l that passes for P and
Q.
II.
For every segment AB and for every segment CD
there exists a unique point E such that B is
between A and E and the segment CD is
congruent to the segment BE.
III. For every point O and every point A not equal to O
there exists a circle with center O and radius OA
IV. All right angles are congruent to each other
V.
For every line l and for every point P that does not
lie on l there exists a unique line m through P that
is parallel to l.
Hilbert’s
Axioms
Incidence
Betweenness
Congruence
Continuity
Parallelism
Note: you need to
read all Chapter
3 while we work
on it.
Every statement
previously
proved in the
text can be used
Notation



By
A*B*C
we will mean “the point B is between the
point A and the point C.”
AXIOMS OF BETWEENNESS (first
part)



B1: If A*B*C then A, B and C are three
distinct points lying on the same line and
C*B*A.
B2: Given two distinct points B and D, there
exist points A, C and E lying on BD such that
A*B*D, B*C*D and B*D*E.
B3: If A, B and C are distinct points lying on
the same line, then one and only one of the
points is between the other two.
EXERCISES



Write the axiom B3 using the notation * we’ve
just introduced.
Can you find a model for the Betweeness
Axioms?
Consider a sphere S in Euclidean threespace and the following interpretation: A point
is a point on S, a line is a great circle on S
and incidence is set membership. Is this
intrepretation a model of Betweeness
Axioms? (What about Incidence Axioms?)
Old definitions revisted


The segment AB is the set of all points C
such that A*C*B together with the points A
and B.
The ray AB is the set of points on the
segment AB together with all the points C
such that A*B*C.
EXERCISE

Let A and B denote two points. Prove that


AB ∩ BA = AB
AB U BA = AB
Definition

Let l be a line. Let A and B be points not lying
on l.


We say that A and B are on the same side of l if
A=B or the segment AB does not intersect l.
We say that A and B are on opposite sides of l
if A ≠ B and the segment AB does intersect l.
Questions

Suppose you have two points A and B lying
on a line l.


Are A and B on the same side of l or on opposite
sides of l?
Suppose you have two points A lying on a
line l and B not lying on l.

Are A and B on the same side of l or on opposite
sides of l?
AXIOMS OF BETWEENNESS (second
part)

B4: For every line l and for every three points
A, B and C not lying on l,


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.
Proposition

If A and B are on opposite sides of l and B
and C are on same side of l then A and C are
on opposite sides of l.
Definition:

A set of points S is a half plane bounded by
a line l if there exists a point A such that S
consists in all the points B for which A and B
are on the same side of l.
Propositions




Every line bounds exactly two half planes
and these two have planes have no point in
common.
If A*B*C and A*C*D then B*C*D and A*B*D.
If A*B*C and B*C*D then A*B*C and A*C*D
(line separation property) If C*A*B and l is
the line through A, B and C then for every
point P lying on l, P lies either on the ray AB
or on the ray AC
Pasch Theorem

If A, B and C are distinct noncollinear points
and l is any line intersecting the line AB in a
point between A and B, then l intersects
either AC or BC. If C does not lie on l then l
does not intersect both AC and BC.
Proposition

If A*B*C then B is the only point lying on the
rays BA and BC and AB=AC.
Definition

A point D is in the interior of an angle <CAB
if


D is on the same side of the line AC as B and
D is on the same side of the line AB as C.
Definition

The interior of a triangle is the intersection
of the interior of its three angles.

A point P is exterior to a triangle if it is not
an interior point of a triangle and does not lie
in any side of the triangle.
Proposition

If D is in the interior of <CAB then



Every point in the ray AD except A is in the interior
of <CAB
None of the points in the ray opposite to the ray
AD are in the interior of <CAB
If C*A*E then B is in the interior of <DAE
Definition

Ray AD is between rays AC and AB if AB
and AC are not opposite rays and D is
interior to <CAB.
Crossbar theorem

If the ray AD is between rays AC and AB then
AD intersects segment BC
EXERCISE (18, Chapter 3) Consider the
following interpretation.


Points: points (x,y) in
the Euclidean plane
such that both
coordinates, x and y,
have the form a/2n
Lines: Lines passing
through several of
those points.
Show that
 The incidence axioms
hold
 The first three
betweenness axioms
hold.
 Line separation property
holds.
 Pasch theorem fail
 What about Crossbar
theorem?