4.2.2 Basic Theorems  Printout
New ideas pass through three periods: It can't be done. It probably can be done, but it's not worth doing. I knew it was a good idea all along!
—Arthur C. Clarke (1917–2008)

        A projective geometry is a non-Euclidean geometry since an immediate result of Axiom 2  is that there are no parallel lines.  (The proofs for Theorems 4.1–4.4 are on the chapters solution page.)

Theorem 4.1. (Dual of Axiom 1) Any two distinct lines are incident with exactly one point.

Exercise 4.1. Write the proof for Theorem 4.1.   

Theorem 4.2. There exist a point and a line that are not incident.

Exercise 4.2. Prove Theorem 4.2. (Caution: You may not assume the existence of any points or any lines.)   

Theorem 4.3. Every line is incident with at least three distinct points.

Exercise 4.3. Prove Theorem 4.3. (Caution: You may not assume a line has any points.)  

Theorem 4.4. Every line is incident with at least four distinct points.

Exercise 4.4. Prove Theorem 4.4. (Hint: You may need to use Axiom 4.)   

Exercise 4.5. Prove the existence of a complete quadrangle.

Exercise 4.6. Is it possible to extend Theorems 4.3 and 4.4 to an arbitrary number of points? Explain.