Fano's Geometry can be described by the following axiom set (slightly different from the one in the text):
1.Every line of the geometry has exactly three points on it.
2.Every pair of distinct points is on exactly one line.
3.Every pair of distinct lines have exactly one point in common.
4.There exists a set of four points, no three of which are on the same line.

The following is a proof of the fact that for any set of three points which are not on the same line, there is a unique fourth point so that these four points satisfy axiom 4. Fill in the numbered blanks.

Pf: Let A,B, and C be a set of three points ____1______. 
By ___2___, each pair of these points is on exactly one line, so the three pairs of points determine exactly three lines. 
By ___3___, each of these lines must contain a third point. 
None of these points can be identical, since otherwise there would exist two lines with two points in common contradicting ___4___.
We have now identified 6 points of the geometry. 
Since there are ___5___ in Fano's geometry (shown in class), there is exactly one additional point, call it F. 
F does not lie on any line containing two of the three original points, so {A,B,C,F} satisfy the property that no three lie on a common line. 
Notice that no other point can be used in place of F.  ❑

Answers:

1. _____not on a common line___________

2. _____Axiom 2 _____________________

3. _____Axiom 1 _____________________

4. _____Axiom 3 _____________________

5. ____7 points _______________________