Incidence Geometry

An incidence geometry is a set \(G\), whose elements are called points, together with a collection of subsets \(\mathcal{L}\), whose elements are called lines, such that the following hold:

(1). For any points \( A,B \in G \), there exists a unique line \( l \in \mathcal{L} \) for which \( A,B \in l \).

(2). For every line \( l \in \mathcal{L} \), \( |l| \geq 2 \).

(3). There exists points \( A,B,C \in G \), and a line \( l \in \mathcal{L} \) such that \( A,B \in l \) but \( C \notin l \).

If \( P \in G \) is a point, and \( l \in \mathcal{L} \) is a line, we say that \( P \) passes, (, or cuts, or lies on) \(l\) provided \( P \in l \).