Plane Geometry - Bruce E. Shapiro

Section 13Incidence GeometryWe will use the expression “a geometry” to refer to the consequences ofa particular set of axioms. For example by Hilbert Geometry we meanthe geometry that is the consequence of Hilbert’s axioms; by EuclideanGeometry we mean the consequences of Euclid’s Axioms, etc. Here we willdescribe a particular type of finite geometry, that is, a geometry witha finite number of points.Incidence Geometry is a term we will use for the geometry that we canderive from the following three axioms.Axiom 13.1 (Incidence Axiom 1) For every pair of distinct points P andQ there exists exactly one line l such that both P and Q lie on l.Axiom 13.2 (Incidence Axiom 2) For every line l there exists at least twodistinct points P and Q such both P and Q lie on l.Axiom 13.3 (Incidence Axiom 3) There exist three points that do not alllie on the same line, i.e., there exists three non-collinear points.Definition 13.4 Three points A, B, C are collinear if there exists at leastone line l such that all three points line on l. They are said to be noncollinearif no such line l exists.Example 13.1 3-Point Plane Geometry. In this example of a finitegeometry that obeys all the axioms of incidence geometry, we define:ˆ A point is an element of the set {A, B, C}ˆ A line is a pair of points such as l = {A, B}ˆ A point P lies on a line l if P ∈ l.53