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Jeremy Gibbs
Modern Geometry
Chapter 2
6.
Proposition 2.3: For every line there is at least one point not lying on it.
Proof:
Given two unique points, P and Q, there exists a unique line PQ which passes through the points by IA 1. Given another distinct point, R, there exists the property that no line is incident with all three of them by IA 3. Thus, for every line, PQ, there will be at least one point, R, not lying on it.
Proposition 2.4: For every point, there is at least one line not passing through it.
Proof:
Given three unique points, P, Q, and R. By IA 1, there exists a unique line, PQ, through P and Q. By IA 3, there exists the property that no line is incident with the above three points, leaving point R not lying on line PQ. Thus, for every point, R, there is at least one line, PQ, that does not pass through it.
Proposition 2.5: For every point P, there exists at least two lines through P.
Proof:
Given three unique points, P, Q, and R. By IA 1, there exists unique lines, PQ, PR, and QR. By IA 3, there exists no line that is incident to all three points. Thus, line PQ is not equal to line PR or line QR, and in similar fashion line PR is not equal to PQ, or QR, and line QR is not equal to line PR or line PQ. As a result of this, there exists two unique lines passing through point P. Thus, for ever point P, there exists at least two lines through P.
9. a. Axioms satisfied: IA 1, IA 2 not satisfied: IA 3 hyperbolic
b. Axioms satisfied: IA 1, IA 2, IA 3 not satisfied: none Euclidean
c. Axioms satisfied: IA 2 not satisfied: IA 1, IA 3 elliptic
d. Axioms satisfied: IA 1, IA 2, IA 3 not satisfied: none hyperbolic
e. Axioms satisfied: IA 2 not satisfied: IA 1, IA 3 elliptic