Euclidean geometry

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nodramaplease

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Which parallel postulate does Fano's geometry satisfy?
Which parallel postulate does the three-point line satisfy?
a) Euclidean parallel postulate
b) Elliptic parallel postulate
c) Hyperbolic parallel postulate

I've already found the axiom's I need. I just don't know which postulate goes with each question.
 
nodramaplease said:
Which parallel postulate does Fano's geometry satisfy?
Which parallel postulate does the three-point line satisfy?
a) Euclidean parallel postulate
b) Elliptic parallel postulate
c) Hyperbolic parallel postulate

I've already found the axiom's I need. I just don't know which postulate goes with each question.
Click to expand...

You are saying that you know the five axioms of Fano's finite geometry.

Do you the definitions of the postulates [a, b &c] referred above?

Can you state those?

Finally can you state the axiom's of Fano's geometry?
 
Axiom 1: Threre exists at least one line.
Axiom 2: There are exactly three points on every line.
Axiom 3: Not all points are on the same line.
Axiom 4: There is exactly on line on any two distict points I.e. through any two points there is a unique line.
Axiom 5: There is at least one point on any two distict lines i.e. Any two lines intersect at least at one point i.e. no two lines can be parallel.

Property 1: For any two distinct points, there is a unique line that is on both of them. Proof: This is Fano axiom 4.
Property 2: Any line has at least two different points on it. Proof: Axiom 2 says "There are exactly three points on every line" Three is at least two.
Property 3: There at least three different points. Proof: Axiom 1 says, "There exists at least one line.
Property 4: Not all points are on the same line. Proof: This is exactly Fano axiom 3.
 
nodramaplease said:
Axiom 1: Threre exists at least one line.
Axiom 2: There are exactly three points on every line.
Axiom 3: Not all points are on the same line.
Axiom 4: There is exactly on line on any two distict points I.e. through any two points there is a unique line.
Axiom 5: There is at least one point on any two distict lines i.e. Any two lines intersect at least at one point i.e. no two lines can be parallel.

Property 1: For any two distinct points, there is a unique line that is on both of them. Proof: This is Fano axiom 4.
Property 2: Any line has at least two different points on it. Proof: Axiom 2 says "There are exactly three points on every line" Three is at least two.
Property 3: There at least three different points. Proof: Axiom 1 says, "There exists at least one line.
Property 4: Not all points are on the same line. Proof: This is exactly Fano axiom 3.
Click to expand...

Do you the definitions of the postulates [a, b &c] referred in your original post?
 
The Euclidean Parallel Postulate. For every line l and for every external point P, there is exactly one line m such that P lies on m and m ||l.
The Elliptic Parallel Postulate: For every line l and for every external point P, there is no line m such that P lies on m and m || l.
The Hyperbolic Parallel Postulate: For every line l and for every external point P, there are at leanst two lines m an such that P lies on both m and n and both m and n are parallel to l.

I think the answers are Hyperbolic for fano's geometry.
and Elliptic for the three point line.

Is that right?
 
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