Beck's Theorem

Beck's Theorem in combinatorial geometry states that for a finite set of points in a plane, either all points lie on a single straight line, or there are a large number of lines that each pass through exactly two of the points. In other words, if the points are not all aligned, then the configuration must generate many different lines determined by pairs of points. This theorem highlights a fundamental relationship between how points are arranged and the complexity of the lines they define, showing that non-collinear point sets create a rich and interconnected geometric structure.