Prove that a Balanced Incomplete Block Design with parameters $(n^2, n^2+n, n+1, n, 1)$ is a finite Affine Plane

Question

I know I need to show that those parameters imply that the following axioms hold when we consider treatments to be points and blocks to be lines:

• A1: given any two points, there is one and only one line that contains them both;
• A2: there is a set of four points, no three of which belong to one common line;
• A3: given any point $p$ and given any line $q$ that does not contain $p$, there is exactly one line that contains $p$ and contains no point of $q$.

Now, $\lambda=1$ implies that given any pair of elements, there is only one block that contains them; i.e. we get A1. I think I have some ideas about how to show A2, but I'm really stumped on how to show that A3 must hold.

Any help would be appreciated!

Answer 1

Let $p$ be a point off the line $q$. The line $q$ has points $a_1,\ldots,a_n$. There are unique lines $l_1,\ldots,l_n$ where $l_j$ passes through $p$ and $a_j$. These lines meet only at $a$. But there are $n+1$ lines through $p$; the last of them does not meet $q$.