**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.

[![enter image description here](https://i.stack.imgur.com/Mq3Op.png)](https://i.stack.imgur.com/Mq3Op.png)

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?