A Finite Geometry Exercise 1.7.
God is like a skillful Geometrician.
—Sir Thomas Browne (1605–1682)

Exercise 1.7. Write the contradiction argument for Fano Theorem 2 which shows that A, B, C, D, E, F, and P are distinct points.

Since D is the third point distinct from A and P on line AP, the point D  is neither A nor P.
Suppose D = B. By Axiom 4, the line AP = DP = BP. Hence {A, B, P} is collinear. Since C is on line AB, Axiom 4 implies that {A, B, C, P} is collinear. But this contradicts that {A, B, C, P} is noncollinear by how P was defined.
The other cases are similar. (Verify the other cases.)