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Lesson Two:
The Axioms of Plane Geometry I
Outline




Undefined terms
Existence and Incidence
Distance
Plane separation
The first set of axioms for plane geometry was proposed by Euclid in his book
“Elements”. By the modern standard, however, his system is not perfect because his
postulations do not contain all the necessary assumptions. Another commonly used
system is proposed by David Hilbert. Hilbert’s axioms which are very much in the spirit
of Euclid’s original work in that they are purely geometry and do not make use of
coordinates or the real numbers. But Hilbert’s system contains too many axioms, we need
to prove many results that appear to be intuitively obvious.
In this part, we present an system of axioms for plane geometry which is based on the
real numbers and measurement. This system was originally introduced by Geoge David
Birkhoff in 1932.
Three major axiom systems for plane geometry
(1) Euclid’s Elements
There are 23 definitions listed . Two of them are
1). A point is that which has no part. 2). A line is breadthless length .
There are five postulations
(2) Hilbert’s Axioms
Undefined terms: point, line, plane, lie, between, congruence
Axioms: There are five groups of axioms
Group I: Axioms of Incidence
Group II: Axioms of order
e.g. If B is between points A and C, then A,B. and C are distinct points on the
same line and B is between C and A.
Group III: Axioms of congruence
e.g. If A and B are two points on a line a and if A’ is a point on the same or
another line a’, then it is always possible to find a point B’ on a given side of the
line a’, such that AB and A' B' are congruent.
Group IV: Axiom of parallel
Group V: Axioms of continuity
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Archimedes axiom. If AB and CD are any segments, then there exists a
number n such that n copies of CD constructed contiguously from A along the
ray will pass beyond the point B.
(3) Birkhoff’s Axioms
Undefined terms: point, line, distance,
Four postulations:
In the following part, we shall list the undefined terms, definitions, axioms and theorems
for plane geometry.
2.1. The undefined terms
In the plane geometry axiom system, the undefined terms are:
point, line, distance, half-plane, angle measure
2.2. Existence and Incidence
Axiom 2.2.1(The Existence Postulate)
The collection of all points forms a nonempty set.
There is more than one point in that set.
Definition 2.2.2 The set of all points is called the plane and is denoted by P.
Axiom 2.2.3(The Incidence Postulate).
Every line is a set of points. For every pair of distinct points A and B there is exactly
one line l containing both A and B.
B
A
l
Notation: The symbol AB is used to denote the line determined by A and B.
Definition 2.2.4 A point A is said to lie on line l, if A is an element of the set l. If a
point B does not lie on the line l, it is called an external point of l.
B
A
l
C
In the above, points A and B lie on l,
and C is an external point of l.
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Definition 2.2.5 Two lines l and m are said to be parallel, written l || m, if there is no
point P that lies on both l and m.
Question: Is every line l parallel to itself?
Why?
Theorem 2.2.6 If l and m are two distinct, nonparallel lines, then there exists exactly
one point P that lies on both l and m.
2.3. Distance
The next axiom tells what we assume regarding the undefined term distance.
Axiom 2.3.1 (The Ruler Postulate)
For every two points A and B there exists a real number
d(A,B),
called the distance from A to B.
For each line l there is a one-to-one correspondence between l and R such that if
A and B are points on the line that correspond to real numbers x and y, respectively,
then d(A,B)=| x-y|.
A
l
0
B
x
y
R
Example 2.3.2 The Cartesian plane
The Caretesian plane is denoted by R2.
Points: (x,y), x and y are real numbers
A line is a set of the form
l={(x,y): x and y are real numbers and ax+by+c=0},
where a and b are fixed real numbers.
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Distance: d (( x1 , y1 ), ( x2 , y2 ))  ( x2  x1 ) 2  ( y2  y1 ) 2 .
B
A
Exercise Find d(A,B), where A=(0,1) B=(-1,-5).
Practice: Find a one-to-one correspondence f between the following line l and R such
that d(A,B)=|f(A)-f(B)| for any two points A and B.
l  {( x, y ) : x  y  1  0}
Definition 2.3.3( Betweeness ). Let A, B and C be three distinct points. The
point C is between A and B, write A  C  B if
C lies on AB and d(A, B)=d(A,C)+d(C,B).
A
Question
Can
C
B
A  C  B and B  A  C both be true?
Definition 2.3.4. Let A and B be two points.
Define the segment AB and the ray AB by :
AB  { A, B}  {C : A  C  B}
AB  AB  {C : A  B  C}
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A
A
B
B
Definition 2.3.5( congruency of line segments). Two segments AB and CD are said to
be congruent, written AB  CD ,
if d(A, B)=d(C, D).
Informally we also say that two segment are equal if they are congruent.
The Ruler Postulate claims that there is a bijection f from each line l to R and
d(A,B)=|f(A)-f(B)| for any two points on the line. Such a function need not be unique.
Definition 2.3.6( Coordinate function).
Let l be a line. A bijection ( or a one-to-one correspondence ) f : l  R is called a coordinate
function for the line l if d(A, B)=|f(A)-f(B)| holds for all points A and B.
The number f(A) is called the coordinate of the point A.
Practice: Show that if f is a coordinate function for a line l, then the function g defined
below is also a coordinate function for l.
g(A)=f(A)+3, for every point A in l.
What general conclusion you can draw from this ?
Theorem 2.3.7.(The Ruler Placement Postulate)
For every pair of distinct points A and B, there is a coordinate function for AB
such that f(A)=0 and f(B)>0.
Proof:
By the Ruler Postulate, there is a one-to-one correspondence f: Let f: AB  R such
that for any points P and Q on AB , d(P,Q)=|f(P)-f(Q)|. Now |f(A)-f(B)|=d(A,B)>0,
assume that f(B)> f(A). Define g: AB  R by g(P)=f(P)-f(A). Then g is a one-to-one
correspondence which is a coordinate function of AB . In addition g(A)=0 and g(B)>0.
Example 2.3.8. The rational plane.
Points are pairs of rational numbers (x,y).
A line in this rational plane is a set of the form
l={(x,y): x and y are rational numbers and ax+by+c=0}
for some fixed rational numbers a,b and c.
For any two points A= ( x1, y1 ) and B  ( x2 , y2 ) , define
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d(A,B)= ( x1  x2 ) 2  ( y1  y2 ) 2 .
(i) The rational plane satisfies the first part of the Ruler Postulate. It also satisfies the
Existence and Incidence Axioms.
(ii) The rational plane does not satisfy the second part of the Ruler Postulate.
(iii) The rational plane satisfies all five postulates posted by Euclidean.
(iv) Consider rational points A=(0,0) and B=(2,0). The circles of radius 2 centered at A and
B do not intersect in the rational plane. In a proof given by Euclidean, he claimed that
such two circles must intersect at a point.
This example shows that Euclidean has used some unstated hypotheses in his
proofs. According to modern standards of rigor, this means that there is a gap in
Euclidean’s proof.
Practice: Show that the two circles in the above example do not intersect in the rational
plane.
2.4. Plane Separation
The next axiom tells us what we need to know about half-plane. This axiom also allows
us to define angle, the interior of an angle, and triangle.
Activity Try to define a triangle. Compare your definition with other’s.
Definition 2.4.1. A set of points S is a convex set if for any two points A and B in S, the
entire segment AB is contained in S.
A
B
Convex
Not Convex
Axiom 2.4.2( The Plane Separation Postulate). For every line l, the points that do not
lie on l form two disjoint, nonempty sets H1 and H 2 , called half-planes bounded by
l, such that the following two conditions are satisfied:
(1). H1 and H 2 are convex.
(2). If P is in H1 and Q is in H 2 , then PQ intersects l.
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H1
H2
Definition 2.4.3. Two rays AB and AC having the same end point are opposite rays if
they are unequal but AB  AC .
C
A
B
Definition 2.4.4( Angle). An angle is the union of two non-opposite rays AB and AC
sharing the same endpoint. The angle is denoted by either BAC or CAB .
The point A is called the vertex of the angle and the rays AB and AC are called the
sides
of the angle.
B
A
C
Remarks
(1) The angle BAC is the same as CAB .
(2) By this definition, the straight angle you learned before is not regarded as an
angle.
Definition 2.4.5. Let A, B and C be three points such that the rays AB and AC are
non-opposite. The interior of angle BAC is defined as follows:
(i) If AB  AC , then the interior is the intersection of the half-plane H B determined
by B and AC and the half-plane determined by C and AB .
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B
A
C
(ii) If AB = AC , then the interior is defined to be empty set.
Discussion
Explain why the definition of interior does not apply to straight angle.
Practice:
Shade the interior of the following angle. Explain your answer.
Definition 2.4.6( Triangle). Let A, B and C be three noncollinear points. The triangle
ABC consists of the three segments AB, BC , AC , that is
ABC  AB  BC  AC .
The points A, B and C are called the vertices of the triangle and the three segments are
called the sides of the triangle.
B
A
C
Practice: Show that that ABC  BAC  ACB .
Theorem 2.4.7 (Pasche’s Axiom). Let ABC be a triangle and let l be a line that does
not contain any of A, B and C. If l intersects AB , then l intersects either BC or
AC .
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A
A
B
B
C
C
Summary
 Undefined terms are
point, line, distance, half-plane, angle measure
 The first four postulate
1. The Existence Postulate
2. The Incidence Postulate
3. The Rule Postulate
4. The Plane Separation Postulate
 Definitions
1. Parallel lines
2. A point between A and B
3. Segment, ray
4. Congruent segments
5. Coordinate function
6. half plane
7. Convex set
8. Angle, interior of an angle
9. Triangle
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Exercise
Set 2
1. Prove the following statements for any points A,B and C.
(1) d(A, B)=d(B, A);
(2) d(A, B)  0 ;
(3) d(A,B)=0 if and only if A=B;
2. Show that for any two points A and B, AB  BA .
3. Assume that f is a coordinate function of the line l.
(1) Show the function g is also a coordinate function for l :
g(A)=f(A)+a, where a is a fixed real number.
(2) Show that if h :l  R is another coordinate function for l, then there must exist a
constant c such that for each point A on l, either h(A)= f(A)+c or h(A)=-f(A)+c.
4.
(1) Find a coordinate function f for the line m below such that f(1,1)=0.
m  {( x, y ) : x  y  2  0}
(2) Let A  (1, 3), B  (0, 2), C  (2, 4) be three points in the Cartesian plane.
Determine which of the following is true:
A  B  C, B  A  C, A  C  B .
(3) Verify that in the Cartesian plane, if A  C  B, then B  C  A.
5. Define  (( x1, y1 ), ( x2 , y2 )) | x1  x2 |  | y1  y2 | for any ( x1, y1 ), ( x2 , y2 ) in the
Cartesian plane.
(1) Show that  is a metric on R 2 , that is
(i )  ( A, B )   ( B, A);
(ii )  ( A, B)  0;
(iii )  ( A, B )  0 iff A  B.
 is called the taxicab metric
(2) If the line l is given by the equation y  mx  b , define f : l  R by
f ( x, y )  x(1 | m |) .
If l is given by the equation x=a, define f by f(a,y)=y.
Verify that the function f is a coordinate function in the taxicab metric.
6. Betweeness in the taxicab metric.
Let  be the taxicab metric for the Cartesian plane.
(a) Draw the sketch of AB for A=(0,0) and B=(0, 1).
(b) Draw a sketch of PQ for P=(0, 1) and Q=(1, 0).
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