Solution Finite Geometry Exercise 1.8.
You know we all became mathematicians for the same reason: we were lazy.
—Maxwell Rosenlicht (1924–1998)

Exercise 1.8. Prove Fano's Theorem 3.

Fano's Theorem 3. Each point in Fano's geometry is incident with exactly three lines.

Proof. Let P be a point. By Axiom 1, there exists a line l. We consider two cases: (1) P is not on l or (2) P is on l.
     Case 1. Assume P is not on line l. By Axiom 2, there are three distinct points on line l, call them A, B, C. Then by Axiom 4, there are three lines AP, BP, and CP. The three lines are distinct; for if the lines were not distinct, Axiom 4 would be contradicted, since P  would be on line l with two of the points A, B, or C. Suppose there is a line k through P. Then by Axiom 5, there is a point D incident with lines l and k. By Axiom 2, D must be either A, B, or C. If D = A, then k = DP = AP. Similarly, for points B and C. Hence, the three lines AP, BP, and CP are the only lines incident with P.
     Case 2. Assume P is on line l. Then by Axiom 2, there are exactly two other points A and B incident with line l. By Axiom 3, there is a point Q not on line l. By Case 1, PQ, AQ, and BQ are the only distinct lines incident with Q and P is not incident with line AQ. Hence, since P is not on line AQ, P is incident with exactly three lines, by Case 1.
     Therefore, by cases 1 and 2, each point is incident with exactly three lines.//