Quiz #2

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.
Below is a proof which shows 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. Provide the justification(either axiom or any theorem concerning Fano's geometry that was talked about in class) for each statement.

Proof:

Let A,B, and C be a set of three points not on a common line.(a) __Assumption / Axiom 4____________
Each pair of these points is on exactly one line, so the three pairs of points determine exactly three lines. (b)__Axiom 2_____________
Each of these lines must contain a third point. (c)____Axiom 1____________
None of these points can be identical, since otherwise there would exist two lines with two points in common. (d) ___Axiom 2 or Axiom 3_______
We have now identified 6 points of the geometry. Since there are 7 points in Fano's geometry, (e)__ Theorem proved in Class (1.8)________________
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.