Incidence geometry: Prove that there is only 1 unique configuration 7_3_

Question

$7_3$ also know as Fano plane. How can I prove that only 1 configuration exists for 7 points and 7 lines with 3 points on every line and 3 lines at every point. I would think an incidence matrix would help, but I'm not sure how to begin. I know a lot of $10_3$ configurations exist, but why only 1 Fano plane configuration. Likewise, I need to figure out how to prove $8_3$ has a unique configuration as well. Any help?

Answer 1

For the Fano plane it is easy just to be systematic.

Choose a line - it contains three points $P_1,P_2,P_3$

Choose a second line having a point in common with the first $P_1,P_4,P_5$

Choose a third line through $P_2$ and $P_4$ - and also therefore $P_6$ (noting that any pair of points defines a distinct line) $P_2,P_4,P_6$.

The fourth line is (necessary) the third line through $P_1$, which must be $P_1,P_6,P_7$ ($P_1$ is already joined to $P_2,P_3,P_4,P_5$)

The fifth line is the third line through $P_2$ which must be $P_2, P_5, P_7$

The sixth line is the third line through $P_4$ which must be $P_3,P_4,P_7$

The seventh line is necessarily $P_3, P_5, P_6$