Axioms
Three-point Plane
Three-point Line
Incidence Geometry
Aaron Cinzori
Fall 2009
Aaron Cinzori
Incidence Geometry
Four-point Geometry
Axioms
Three-point Plane
Three-point Line
Undefined Terms and Axioms
Undefined terms: point, line, lie on.
Aaron Cinzori
Incidence Geometry
Four-point Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Undefined Terms and Axioms
Undefined terms: point, line, lie on.
Incidence Axiom 1: For every pair of distinct points P and Q,
there exists exactly one line ` such that both P and
Q lie on `.
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Undefined Terms and Axioms
Undefined terms: point, line, lie on.
Incidence Axiom 1: For every pair of distinct points P and Q,
there exists exactly one line ` such that both P and
Q lie on `.
Incidence Axiom 2: For every line ` there exist at least two distinct
points P and Q such that both P and Q lie on `.
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Undefined Terms and Axioms
Undefined terms: point, line, lie on.
Incidence Axiom 1: For every pair of distinct points P and Q,
there exists exactly one line ` such that both P and
Q lie on `.
Incidence Axiom 2: For every line ` there exist at least two distinct
points P and Q such that both P and Q lie on `.
Incidence Axiom 3: There exist three points that do not all lie on
any one line.
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Undefined Terms and Axioms
Undefined terms: point, line, lie on.
Incidence Axiom 1: For every pair of distinct points P and Q,
there exists exactly one line ` such that both P and
Q lie on `.
Incidence Axiom 2: For every line ` there exist at least two distinct
points P and Q such that both P and Q lie on `.
Incidence Axiom 3: There exist three points that do not all lie on
any one line.
Definition: Three points A, B and C are collinear if there exists
one line ` such that all three of the points A, B, and
C all lie on `. The points are noncollinear if there is
no such line `.
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Three-point Plane
Interpret point to mean one of the three symbols A, B, C .
Interpret line to mean an unordered pair of points.
Interpret lie on to mean “is an element of”.
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Three-point Plane
Interpret point to mean one of the three symbols A, B, C .
Interpret line to mean an unordered pair of points.
Interpret lie on to mean “is an element of”.
How many lines are there are there in this interpretation?
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Three-point Plane
Interpret point to mean one of the three symbols A, B, C .
Interpret line to mean an unordered pair of points.
Interpret lie on to mean “is an element of”.
How many lines are there are there in this interpretation?
Is this a model for incidence geometry?
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Three-point Line
Interpret point to mean one of the three symbols A, B, C .
Interpret line to mean the set of all points points.
Interpret lie on to mean “is an element of”.
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Three-point Line
Interpret point to mean one of the three symbols A, B, C .
Interpret line to mean the set of all points points.
Interpret lie on to mean “is an element of”.
How many lines are there are there in this interpretation?
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Three-point Line
Interpret point to mean one of the three symbols A, B, C .
Interpret line to mean the set of all points points.
Interpret lie on to mean “is an element of”.
How many lines are there are there in this interpretation?
Is this a model for incidence geometry?
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Four-point Geometry
Interpret point to mean one of the four symbols A, B, C , D.
Interpret line to mean an unordered pair of points.
Interpret lie on to mean “is an element of”.
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Four-point Geometry
Interpret point to mean one of the four symbols A, B, C , D.
Interpret line to mean an unordered pair of points.
Interpret lie on to mean “is an element of”.
How many lines are there are there in this interpretation?
Aaron Cinzori
Incidence Geometry
Axioms
Three-point Plane
Three-point Line
Four-point Geometry
Four-point Geometry
Interpret point to mean one of the four symbols A, B, C , D.
Interpret line to mean an unordered pair of points.
Interpret lie on to mean “is an element of”.
How many lines are there are there in this interpretation?
Is this a model for incidence geometry?
Aaron Cinzori
Incidence Geometry