Incidence GeometrySeptember 22, 2009c 2009 Charles DelmanWhat is Incidence Geometry?v Incidence geometry is geometry involving only points, lines, and theincidence relation. It ignores the relations of betweenness and congruence.v Incidence geometry based on the three neutral incidence axioms mightbe termed neutral incidence geometry.v These axioms might be modified or expanded to obtain special types ofincidence geometry.c 2009 Charles Delman 1DefinitionsRecall the following definitions based on the relation of incidence:v A set of points is collinear if there is a common line incident with all ofthem. A set of points that is not collinear is called noncollinear.v A set of lines is concurrent if there is a common point incident with allof them.v Two lines are parallel if there is no point incident with both of them.(That is, they do not intersect; in this case, the word “intersect” isgiven a meaning distinct from its set-theoretic one, since lines are notnecessarily considered to be sets of points.) More generally, a set oflines is parallel if, for any two distinct lines in the set, there is no pointincident with both of them.c 2009 Charles Delman 2The Axioms of Neutral Incidence GeometryRecall the three neutral incidence axioms:v Axiom I-1: For every point P and for every point Q that is distinct fromP , there is a unique line l incident with P and Q. (That is, t here is aline incident with both P and Q, and if lines l and m are each incidentwith both P and Q, then l = m.)v Axiom I-2: For every line l there exist (at least) two points incidentwith l.v Axiom I-3: There exist three distinct noncollinear points.c 2009 Charles Delman 3Recall also the following notation:v The unique line incident with points P and Q is denoted by←→P Q.c 2009 Charles Delman 4Propositions of Neutral Incidence GeometryIt is instructive and useful to establish some theorems that can be provenusing only these three axioms. In fact, for the following five propositions, wewill not even need to use Axiom I-2! Of course, these theorems remain true(and proven) when more axioms are added. The propositions are numberedin accordance with the textbook.v Proposition 2.1 If l and m are distinct lines that are not parallel, thenl and m have a unique point in comm on.To get a feel for this proposition, let us consider an equivalent statement,its contrapos itive.v Proposition 2.1 If l and m do not have a unique point in common, thenl and m are parallel or l = m.c 2009 Charles Delman 5A Proof of Proposition 2.1Proof. [This proof is based on the first statement: If l and m are distinctlines that are not parallel, then l and m have a unique point in common.]Let l be a line, and let m 6= l be a line that is not parallel to l. [Here weassume the conditions of the theorem. Note that l can be any line, butthen restrictions depending on l apply to our choice of m.]Claim 1. Lines l and m have a point in common. The claim followsimmediately from the definition of parallel.Claim 2. Lines l and m do not have two distinct points in common. LetP be a point they have in common. [Choosing such a point P is justifiedby Claim 1.] By way of contradiction, suppose Q lies on both l and mand Q 6= P . [Note that this is really an exis tence statement: there existsa point Q lying on both l and m such that Q 6= P . Alternatively, wecould have let Q be a point on both l and m (which we know exists -c 2009 Charles Delman 6the set of such points is not empty, by Claim 1) and supposed only thatQ 6= P . Then we would be proving (still by contradiction) the equivalentuniversal statement that, for all points Q lying on both l and m, Q = P .In general, (6 ∃a ∈ A)p(a) is logically equivalent to (∀a ∈ A)¬p(a). ]Since Q 6= P , it follows by the uniqueness c onclusion of Axiom I-1 thatl = m. This contradicts our initial assumption that l 6= m; therefore, weconclude there is no such point Q, so P is unique.c 2009 Charles Delman 7To summarize, without the commentary, we have:Proof. Let l be a line, and let m 6= l be a line that is not parallel to l.Claim 1. Lines l and m have a point in common. The claim followsimmediately from the definition of parallel.Claim 2. Lines l and m do not have two distinct points in common. LetP be a point they have in common. By way of contradiction, supposeQ lies on both l and m and Q 6= P . Since Q 6= P , it follows by theuniqueness conclusion of Axiom I-1 that l = m. This contradicts ourinitial assumption that l 6= m; there fore, we conclude there is no suchpoint Q, so P is unique.c 2009 Charles Delman 8Another proof of Proposition 2.1Proof. [This proof is based on the second statement: If l and m do nothave a unique point in comm on, then l and m are parallel or l = m.]Assume it is not the case that l and m have a unique point in common.Then either they have no point in common or there exist two distinctpoints that they have in common.Case 1: l and m have no point in common. Then, by definition, they areparallel.Case 2: There are two distinct points common to l and m. Then by theuniqueness conclusion of Axiom I-1, l = m.c 2009 Charles Delman 9v Proposition 2.2 There exist three distinct lines that are not concurrent.v Proposition 2.3 For every line, there is at least one point not incidentwith it.v Proposition 2.4 For every point, there is at least one line not incidentwith it.v Proposition 2.5 For every point, there are at least two lines incidentwith it.c 2009 Charles Delman 10Parallelism Propertiesv Now that we have seen some models of incidence geome try, we see thatdifferent properties with respect to parallel lines may apply. The sameproperty need not even apply to all the lines and points of a given m odel.However, geometries with uniform properties are generally of greaterinterest, and the following three possibilities are most important.© If l is a line and P is a point not lying on l, then there is a unique linethrough P that is parallel to l. (Euclidean Parallel Property)© If l is a line and P is a point not lying on l, then there exist twodistinct lines through P that are parallel to l. (Hyperbolic ParallelProperty)© Any two distinct lines are incident with a common point. (That is,if l is a line and P is a point not lying on l, then there are no linesthrough P that are parallel to l. (Elliptic Parallel Property)c 2009 Charles Delman 11Affine and Projective Geometryv Incidence geometry with the additional assumption of the EuclideanParallel Property is