Fano plane line I am confused on this question I know what a fano line is where it contains exactly 3 points.
If we are given 2 of three points how could we find the third one? this question is on my head for so long but cant seem to figure out.
thank you
 A: This probably won't help, but it connects with more general theory. The Fano plane consists of all points $(a,b,c)$, where $a$, $b$ and $c$ are $0$ or $1$, and $(0,0,0)$ is not allowed. Given two points in this notation, we obtain the third point on the line by adding the coordinates modulo $2$ (so $1+1=0$). 
Now go to the picture that you were given a link to. The labels there have been chosen to be consistent with the "binary" description given in the previous paragraph. Look for example at the points they call $3$, $5$, and $6$, and that I would call $(0,1,1)$, $(1,0,1)$, and $(1,1,0)$.
Find for example $(0,1,1)+(1,0,1)$ modulo $2$. We get $(1,1,0)$!  It is the same with all the others. To find the third point on the line, given that the coordinates of two of the points are $(a,b,c)$ and $(d,e,f)$, add coordinate-wise modulo $2$. 
A: see this picture:
http://en.wikipedia.org/wiki/File:Fanoperm364.svg
Given any two points you can find the third.
in fact the set of lines are:
$$\mathcal B= \{\{1,2,3\},\{1,4,5\},\{1,6,7\},\{3,4,7\},\{2,4,6\},\{2,5,7\},\{3,6,5\}\}$$
if $X=\{1,2,...,7\}$, then $(X,\mathcal B)$ is a 2-(7,3,1)-design  (a steiner system in fact).
A: Note that the Fano plane is an incidence structure in which any two points are contained in exactly one line. And also, each line contains exactly 3 points. Therefore, if you are given any two points there is a unique line containing them and thus you can figure out what the third point is.
A picture might be helpful here, see the wikipedia page for a depiction of the Fano plane.
For example, if you were given points $3$ and $5$. Then by looking at the picture you can see that the unique line containing $3$ and $5$ is $\{3,5,6\}$ and therefore the third point would be $6$.
