Problem: Show that any two points in Fano plane are not contained in exactly two lines of the plane and their sum is contained in those two lines in which $p$ and $q$ are not contained.
My attempt:
For the proof we will use Homogeneous coordinates, that is triples with elements of the field $\mathbb{F}_2$. We will construct Fano plane in such a way that for any two points $p$ and $q$ the third point on the line has the label formed by adding the labels of $p$ and $q$ modulo 2.
Take any two arbitrary points $p$ and $q$, which are not equal. Then $p,q \in \mathbb{F}_2^{3*}$. Then $p,q$ are both contained in a line where the third point is the sum of $p$ and $q$. Hence on each line of the Fano plane there are two linearly independent points and third one is formed in a way that it is the sum of the linearly independent points.
I am stuck right now could anyone please explain or give hints?
Why each point is contained in exactly three lines?
Best Answer
Andreas Caranti already explained why each point belongs to exactly three lines. The other question can be settled as follows.
Let $p$ and $q$ be the two points in question. The third point on the line determined by $p$ and $q$ is thus $p+q$. We know that $p+q$ is on exactly three lines. Call them $L_1$, $L_2$ and $L_3$. Without loss of generality $L_1=\{p,q,p+q\}$.
The point $p$ cannot belong to either $L_2$ or $L_3$ for then there would be two lines containing both $p$ and $p+q$. Similarly we see that $q$ cannot be on either $L_2$ or $L_3$ either.
Recall that there are $7$ lines. The point $p$ lies on exactly three of them - two others in addition to $L_1$. By the above observation these two lines are previously unnamed, so we choose to call them $L_4$ and $L_5$. Observe that neither of them can contain $q$. Therefore the lines containing $q$ are $L_1$ and the two yet unnamed lines $L_6$ and $L_7$.
We have covered all the seven lines, and our census gives the conclusion: $L_2$ and $L_3$ are the only lines not containing either $p$ or $q$. As required, those were also the two lines passing thru $p+q$ (excluding $L_1$).