1. Axiomatic System
 Undefined Terms
 Technical Terms : Definitions
defined within these Undefined Terms
 Set of statements(Axioms)
dealing with Undefined Terms and definitions
 All other Statements of the system- Theorems
Logical consequences of these axioms.

2. Independent
An axiom is said to be independent
If it can not be logically deduced from the
other axioms in the system.

3. Completeness
A set of axioms is said to be complete
If it is not possible to add any
independent axiom on the system.

4. Consistency
A set of axioms is said to be consistent
If it is impossible to deduce a theorem
(from axioms) that contradict any axiom
or previously proved theorem.

5. Euclidean Geometry
 The Euclidean Geometry is an axiomatic system along with five axioms .
The five axioms are:
1. To draw a straight line from any point to any point.
2. To produce [extend] a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance [radius].
4. That all right angles are equal to one another.
5. The parallel postulate: That, if a straight line falling on two straight lines
make the interior angles on the same side less than two right angles, the
two straight lines, if produced indefinitely, meet on that side on which are
the angles less than the two right angles.
It is possible to reject any or all of these and still have valid geometries with different
principles. For example, if we simply toss out #5, we are no longer dealing with
Euclidean Geometry, we are dealing with Hyperbolic geometry (in which there are
infinite distinct lines parallel to another line through a given point)…. And if …

6. Geometry
Euclidean Geometry Neutral Geometry Non-Euclidean Geometry
there are infinite
distinct lines parallel
Axiom:-5 through a given point
there is unique line
Common
parallel through a
given point
Axioms
…… Axiom:-5
1,2,3,4 there are no lines
parallel through a
given point

7. Four Point Geometry
Undefined Terms: Point, Line, On.
Axiom 1: There exist exactly four distinct points.
Axiom 2: Any two distinct points have exactly one line on both.
Axiom 3: Each line is on exactly two points.
Def 1: Two lines on the same point are said to intersect
Def 2: Two lines that do not intersect are called parallel.
Theorem 1: Each point of the four-point geometry has exactly three lines on it
…

8. Are blue lines parallel?
Complete four point
Yes, see definition 2

9. Complete four point
Fano-Configuration