Joseph Fagyas
Texas A&M University
MATH 482
Instructor: Dr. David Larson
Final Report: An Introduction to Neutral Geometry
When deriving proofs, there are some general proof requirements that we must accept
before starting the proof. The requirements needed are: a mutual understanding of the meaning
of words and/or symbols used, acceptance of axioms/postulates without further justification, and
agreement on logic of proofs (i.e. how and when one statement follows another). Once we accept
these requirements, we may move forward with our proof.
There are five primitive geometric terms used as a basis for defining all other geometric
terms (in plane geometry). These words are point, line, incidence, between, and congruent. We
will consider these terms to be “undefined.” The reason they are technically undefined is that
they may not always mean the exact same thing in different geometries/contexts. However, we
will use the conventional definitions of the words for our practices.
The beginnings of geometry go back thousands of years, to the times of ancient Greece.
One particular mathematician, Euclid, was responsible for many contributions to geometry.
Elements was Euclid’s compilation of 13 volumes of Greek plane and solid geometry, proportion
theory, number theory, definitions, postulates, propositions, and proofs. One major contribution
that Euclid had was his five geometric postulates that, over the years, mathematicians have
investigated and tried to prove. These postulates were as follows (though none of Euclid’s
original texts survived, so translations may vary):
1. For every point P and every point Q not equal to P there exists a unique line that
passes through P and Q.
2. For every segment AB and every segment CD there exists a unique point E on
line AB such that B is between A and E and segment C is congruent to segment
BE.
3. For every point O and point A not equal to O there is a circle with center O and
radius OA.
4. All right angles are congruent (to each other).
5. If a line intersects two straight lines forming two interior angles on the same side
that sum to less than two right angles then the two lines, if extended indefinitely,
meet on that same side (on which the angles sum to less than two right angles).
More recently than Euclid though, was David Hilbert (1862-1943), who established 14
axioms which can be used to define neutral geometry. This is a geometry based on the undefined
terms listed above. It is “neutral” in the sense that it does not have a parallel postulate, which we
will later see will define the types of geometry past neutral geometry. There are four categories
of axioms in neutral geometry: incidence, betweenness, congruence, and continuity. Note: a
segment will be defined as points A≠B and all points on line AB between A and B.
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The first set of axioms we will study is the set of “incidence” axioms. These have to do with
points, lines, and their relationship to one another. They are as follows:
1. For every point P and every point Q not equal to P, there exists unique line l with
P and Q incident to l.
2. Every line l has at least two distinct (unique) points incident to it.
3. There exist three distinct points with the property that there is no line with all
three points incident to it.
The second set of axioms in neutral geometry is the set of “betweenness” axioms. These
axioms describe points in relation to one another, both on and off lines. Note: The symbol * will
be used to denote betweeness (i.e. A*B*C means B is between A and C)
1. If A*B*C then A, B, and C are three distinct collinear points and C*B*A.
2. Given 2 distinct points B and D (B≠D) then there exist points A, C, and E lying
on line BD such that A*B*D, B*C*D, and B*D*E.
3. If A, B, and C are three distinct collinear points then exactly one is between the
other two, i.e. A*B*C or A*C*B or B*A*C.
4. For all lines l and every set of three points A, B, and C not on l:
i) If A and B are on the same side of l and if 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 if B and C are on opposite sides of
l, then A and C are on the same side of l.
Some propositions and a theorem that result from the previous axioms include, but are
not limited to, the following:
 Proposition: Given A*B*C and A*C*D then B*C*D and A*B*D.
 Proposition: A*B*C implies that B is the only point common to rays BA and BC
and that ray AB = ray BC.
 Pasch’s Theorem: If A, B, and C are three distinct collinear points and l is a line
meeting segment AB at a point between A and B then l also meets segment AC or
segment BC, and meets both only if it goes through the point C.
Proof. Case 1. Is C on l? If so, we are done (conditions satisfied)
Case 2. C is not on l. This implies that A and B are on opposite sides of l
because segment AB meets l at a point between A and B (by definition)
and that C is either on the same side of l as A or same side of l as B.
Case 2a) C and A are on the same side. A and B are on opposite
sides, by definition. This implies that C and B are on opposite sides
of l and so l meets segment CB.
Case 2b) C and B are on the same side. A and B are on opposite
sides, by definition. This implies that C and A are on opposite
sides of l and so l meets segment CA.
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The third set of axioms that we will investigate is the set of “congruence” axioms. These
axioms tell us how to determine if segments and angles are perfectly alike one another. Note:
The symbol ≅ will be used to denote congruence. The symbol ∢ will be used to denote an angle
(i.e. ∢ABC means angle ABC). And the symbol ∆ will be used to denote a triangle (i.e. ∆ABC
means triangle ABC). These axioms are as follows:
1. If A and B are two distinct points and if A’ is any point, then for each ray r
emanating from A’ there exists a unique point B’ on r with B’≠A’ and segment
AB≅A’B’.
2. If segment AB≅CD and segment AB≅EF, then CD≅EF. Moreover, every
segment is congruent to itself.
3. If A*B*C and A’*B’*C’ and AB≅A’B’ and BC≅B’C’, then AC≅A’C’.
4. Given any angle ∢CAB and any ray A’B’, then there exists a unique ray A’C’ on
a given side of line AB such that ∢CAB≅∢C’A’B’.
5. If ∢A≅∢B and ∢A≅∢C, then ∢B≅∢C. Moreover, every angle is congruent to
itself.
6. (SAS) If two sides and the included angle of one triangle are congruent to the two
corresponding sides and included angle of another triangle, then the two triangles
are congruent.
Some propositions and a theorem we exhibit from the previous axioms include, but are
not limited to, the following:
 Proposition: Supplements of congruent angles are congruent.
 Proposition: a) Vertical angles are congruent.
b) An angle congruent to a right angle is a right angle
 Proposition: (ASA) Given ∆ABC and ∆DEF with ∢A≅∢D, ∢C≅∢F, and
segment AC≅DF, then the triangles are congruent.
 Proposition: (SSS) Given ∆ABC and ∆DEF, with AB≅DE, AC≅DF, and
BC≅EF, then ∆ABC≅∆DEF.
 Alternate Interior Angle Theorem: Let t be a line transversal to two lines l and l’,
at two different points, with a pair of congruent alternate interior angles. Then l
and l’ are parallel.*
*NOTE: This theorem does not work backwards. i.e. Lines l and l’ being parallel does not imply that the
alternate interior angles are congruent.
The final axiom of neutral geometry we will examine is the “continuity” axiom. This
axiom essentially asserts that a line is a continuous set of points, made up of two rays and one
shared point, or vertex. The continuity axiom is as follows:
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1. Suppose the set {l} of all point incident to a line l is a disjoint union of two
nonempty sets S1 and S2. Furthermore, no point in S1 is between any two points in
S2, and likewise, no point in S2 is between any two points in S1. Then there is a
unique point P on l such that one subset is a ray with vertex P and the other is the
opposite ray (less P).
Symbolic notation for this axiom:
S1 ≠∅, S2 ≠∅, {l} = S1∪S2, and ∅= S1∩S2
Now that we have established a foundation for neutral geometry, we can look past this
geometry to others that, with the presence of a parallel axiom, are often used in the world
mathematics. Neutral geometry, of course, is a basis for geometry; however, once we introduce a
parallel axiom, we move from neutral geometry to a “higher” geometry. We may only introduce
one parallel axiom to our geometry though because only one can be true at a time. A few
examples include Euclidean (which is the geometry most of us are familiar with), Hyperbolic,
and Elliptic. The following are the parallel axioms for these geometries, respectively.
The first axiom is the Euclidean parallel axiom and is often referred to as “Hilbert’s
Axiom of Parallelism.” Adding this axiom to our list of neutral geometry axioms puts us in the
geometry we are all most familiar with, Euclidean. The axiom is as follows: For every line l and
every point P not on l, there is at most one line m with point P on m and m parallel to l.
The second axiom is the hyperbolic parallel axiom and is the negation of Hilbert’s
Axiom. This axiom is as follows: There exist a line l and a point P not on l with two or more
lines m and m’ (with m≠m’) through P parallel to l.
Neutral geometry builds a foundation for other geometries and lets us better understand
the most basic mechanics behind them. However, we do not stop at neutral geometry. We can go
on to Euclidean, hyperbolic, and other geometries- each of which offers different perspectives
and results from the next.
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Works Cited
Greenberg, Marvin J. Euclidean and non-euclidean geometries. 4th ed. W. H. Freeman and
Company, 2008. Print.
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