Neutral Geometry October 25, 2009 c 2009 Charles Delman
Taking Stock: where we have been; where we are going Set Theory & Logic Terms of Geometry: points, lines, incidence, betweenness, congruence. Incidence Axioms Propositions of Incidence Geometry Betweenness Axioms T h eo r e m s Congruence Axioms T h eo r e m s Philosophical Issues Models Affine & Projective Geometry c 2009 Charles Delman 1
In neutral geometry we assume the first three incidence axioms Axiom I-1: For every point P and for every point Q that is distinct from P, there is a unique line l incident with P and Q. Axiom I-2: For every line l there exist (at least) two points incident with l. Axiom I-3: There exist three distinct noncollinear points. Remark. Any additional assumption made about the existence of parallel lines, such as the elliptic, Euclidean, or hyperbolic parallel property would also be an incidence axiom. That we are not making any such assumption is the reason the theory we are currently studying is called neutral. It is neutral on the question of parallelism. c 2009 Charles Delman 2
Using only the three axioms above (in fact, using only I-1 and I-3), we have so far proven the following propositions: Proposition 2.1 If l and m are distinct lines that are not parallel, then l and m have a unique point in common. Proposition 2.2 There exist three distinct lines that are not concurrent. Proposition 2.3 For every line, there is at least one point not incident with it. Proposition 2.4 For every point, there is at least one line not incident with it. Proposition 2.5 For every point, there are at least two lines incident with it. c 2009 Charles Delman 3
Betweenness Axioms Recall that P Q R means that point Q is between points P and R. Axiom B-1: If P Q R, then P, Q, and R are distinct, collinear points, and R Q P. Axiom B-2: Given points Q and S, there are points P, R, and T incident with line QS such that P Q S, Q R S, and Q S T. Remark. Axiom B-2 does not assert P Q R or any other relationship among the points that does not involve both Q and S. The obvious additional relationships are true but must (and will) be proven! Axiom B-3: If P, Q, and R are distinct collinear points, then one and only one of them is between the other two. c 2009 Charles Delman 4
These three axioms suffice to prove: Proposition 3.1 Let A and B be distinct points. Then (i) AB BA = AB, and (ii) AB BA = { AB} (where { AB} denotes the set of points lying on line AB). It will be useful to first prove the following lemmas: Lemma 1. Let A and B be distinct points. Segment AB is equal to segment BA. Lemma 2. Let A and B be distinct points. The following three sets of points on line AB are disjoint: {P P : P A B}, segment AB, {P P : A B P }. c 2009 Charles Delman 5
Sides of a Line To formulate the last axiom of betweenness, we need a definition. Definition: Given a line l and two distinct points P and Q that are not incident with l, P and Q are on the same side of line l if... (Exercise!). c 2009 Charles Delman 6
Definition: Given a line l and two points P and Q that are not incident with l, P and Q are on the same side of line l if P = Q or if no point in segment P Q is incident with l. If P and Q are not on the same side of l, they are said to be on opposite sides of l. Axiom B-4 (Plane Separation) For every line l: (i) The relation of being on the same side of l is transitive. (ii) For any three points P, Q, and R not lying on l, if P and Q are on opposite sides of l and Q and R are on opposite sides of l, the P and R are on the same side of l. Corollary (iii). If P and Q are on opposite sides of l and Q and R are on the same side of l, then P and R are on opposite sides of l. Lemma 3. The relation of being on the same side is an equivalence relation. c 2009 Charles Delman 7
What Axiom B-4 Means Intuitively (i) Statement (i) ensures that our geometry is two-dimensional: Q P R c 2009 Charles Delman 8
What Axiom B-4 Means Intuitively (ii) Statement (ii) ensures there are no singular lines at which more than two half-planes meet. Q P R c 2009 Charles Delman 9
Half-planes Notation. Let [P ] l denote the equivalence class of P. This notation, different from that used in the text, recognizes that each distinct line determines a different equivalence relation. The equivalence class [P ] l is often called the half-plane bounded by l that contains P. (The text uses H P to denote this half-plane, with the specific line l bounding the half-plane understood from context.) Proposition 3.2 Every line bounds exactly two distinct (and disjoint) half-planes The fact that distinct half-planes are disjoint follows from Lemma 3; distinct equivalence classes are always disjoint. It remains to prove that there are exactly two. c 2009 Charles Delman 10
Review of Betweenness Axioms, Definitions, & Propositions Axiom B-1: If P Q R, then P, Q, and R are distinct, collinear points, and R Q P. Axiom B-2: Given points Q and S, there are points P, R, and T incident with line QS such that P Q S, Q R S, and Q S T. Axiom B-3: If P, Q, and R are distinct collinear points, then one and only one of them is between the other two. Proposition 3.1 Let A and B be distinct points. Then (i) AB BA = AB, and (ii) AB BA = { AB}. c 2009 Charles Delman 11
Definition: Given a line l and two points P and Q that are not incident with l, P and Q are on the same side of line l if P = Q or if no point in segment P Q is incident with l. If P and Q are not on the same side of l, they are said to be on opposite sides of l. Axiom B-4 For every line l: (i) The relation of being on the same side of l is transitive. (ii) For any three points P, Q, and R not lying on l, if P and Q are on opposite sides of l and Q and R are on opposite sides of l, the P and R are on the same side of l. Corollary (iii). If P and Q are on opposite sides of l and Q and R are on the same side of l, then P and R are on opposite sides of l. Proposition 3.2 Every line bounds exactly two distinct (and disjoint) half-planes). c 2009 Charles Delman 12
Proposition 3.3 Given A B C and A C D, then B C D and A B D. Corollary. Given A B C and B C D, then A B D and A C D. Proposition 3.4 Given A B C, let P be a point on the line (given by Axiom B-1) through A, B, and C. Then P BA or P BC. Pasch s Theorem Given ABC, if line l intersects side AB, then it intersects either side AC or side BC. Furthermore, unless l passes through C, it intersects only one of these two sides. Remark. Pasch s Theorem is equivalent to Axiom B-4, and is sometimes taken as an axiom instead. Proposition 3.5 If A B C, then AB BC = AC and AB BC = {B}. c 2009 Charles Delman 13
Proposition 3.6 Given A B C. (i) AB = AC. (ii) BA BC = B. Remark. The seemingly strange grouping of statements in this proposition is due to the fact the part (i) is useful for proving part (ii), since it follows from (i) that BA BC CA AC = AC (Proposition 3.1). One can then apply Proposition 3.5. Corollary. If point P lies on ray AB, then AP = AB. Corollary. A ray has one and only one opposite ray. c 2009 Charles Delman 14
Definition: The interior of angle CAB is the set of points D such that D is on the same side of line AB as C and also on the same side of line AC as B. C A B Definition: The interior of triangle CAB is the intersection of the interiors of angles A, B, and C. Remark. It is easy to prove that if a point is in the intersection of the interiors of two angles of a triangle, then it is in the interior of the third. c 2009 Charles Delman 15
Proposition 3.7 Given BAC and a point D on line BC, then D is in the interior of BAC if and only if B D C. Remark. It is possible that a point in the interior of an angle does not lie on any line intersecting both sides of the angle. If the Euclidean Parallel Property is assumed, then this situation is precluded (why?), but not otherwise. c 2009 Charles Delman 16
Lemma 4. If point A lies on line l and point B does not lie on line l, then every point of ray AB, other than A, lies on the same side of l. Proposition 3.8 Given point D in the interior of BAC. Every point of ray AD is in the interior of BAC. No point on the opposite ray to AD is in the interior of BAC. If C A E, then B is in the interior of DAE. Definition: Ray AC is between rays AB and AD if AB and AD are not opposite rays, and C is in the interior of BAD. Remark. This is a different usage of the word between from the undefined term referring to points. Remark. The first part of Proposition 3.8 is required to justify this definition. (Why?) c 2009 Charles Delman 17
Definition: The exterior of a triangle is the complement of the union of its interior and sides. Proposition 3.9 If a ray emanates from an exterior point of a triangle and intersects one of the sides, then it also intersects one of the other two sides. If a ray emanates from an interior point of a triangle, then it intersects one of the sides; if it does not contain a vertex, then it intersects only one side. How does the first part of Proposition 3.9 differ from Pasch s Theorem? c 2009 Charles Delman 18
Congruence Axioms The word congruence has two distinct uses, both undefined. One refers to a relation among segments, the other two a relation among angles. The first three congruence axioms refer to congruence between segments. Axiom C-1: Given distinct points A and B, any point C, and any ray r emanating from C,! point D on ray r such that AB = CD. Axiom C-2: Congruence among segments is an equivalence relation. Remark. Note the clever way that transitivity is described in the text so that symmetry may be deduced from reflexivity and transitivity! Axiom C-3: If A B C, D E F, AB = DE and BC = EF, then AC = DF. c 2009 Charles Delman 19
The next two congruence axioms refer to congruence between angles and are analogues of Axioms C-1 and C-2. We do not need an analogue of Axiom C-3 for angles, because it can be proven. Axiom C-4: Given any angle BAC, any ray DE, and a chosen side of line DE,! ray DF on the given side of DE such that BAC = EDF. Axiom C-5: Congruence among angles is an equivalence relation. The final congruence axiom refers to both types of congruence and is the most powerful of all. In order to state it, we must define yet another relation called congruence in terms of congruence for segments and angles! c 2009 Charles Delman 20
Congruence Among Triangles Definition: Two triangles are congruent if there is a (1-1, onto) correspondence between their sets of vertices such that the corresponding angles and sides are all congruent. Notation. ABC = DEF means that A = D, B = E, C = F, AB = DE, BC = EF, and AC = DF. Thus the word congruence has three uses: an undefined relation on the set of segments; an undefined relation on the set of angles; and a relation on the set of triangles defined in terms of these two undefined relations. Axiom C-6 (SAS Criterion for Triangle Congruence): Given triangles ABC and DEF, if AB = DE, BC = EF, and B = E, then ABC = DEF. c 2009 Charles Delman 21
The SAS criterion tells us our geometry is homogeneous: it looks the same from any point. c 2009 Charles Delman 22