possible confusion with the terminology of the classical case, the lines are referred to as cycles or blocks. Specifically, a Möbius plane is an incidence structure of points and cycles such that: Every triple of distinct points is incident with precisely one cycle. For any flag and any point not incident with there is a unique cycle with and . (The cycles are said to touch at .) Every cycle has at least three points and there exists at least one cycle. The incidence structure obtained at any point of a Möbius plane by taking as points all the points other than and as lines only those cycles that contain (with removed), is an affine plane. This structure is called the residual at in design theory. A finite Möbius plane of order is a tactical configuration with points per cycle that is a 3-design, specifically a block design. Incidence theorems in the Euclidean plane The Sylvester-Gallai theorem A question raised by J.J. Sylvester in 1893 and finally settled by Tibor Gallai concerned incidences of a finite set of points in the Euclidean plane. Theorem (Sylvester-Gallai): A finite set of points in the Euclidean plane is either collinear or there exists a line incident with exactly two of the points. A line containing exactly two of the points is called an ordinary line in this context. Sylvester was probably led to the question while pondering about the embeddability of the Hesse configuration. The de Bruijn–Erdős theorem A related result is the de Bruijn–Erdős theorem. Nicolaas Govert de Bruijn and Paul Erdős proved the result in the more general setting of projective planes, but it still holds in the Euclidean plane. The theorem is: In a projective plane, every non-collinear set of points determines at least distinct lines. As the authors pointed out, since their proof was combinatorial, the result holds in a larger setting, in fact in any incidence geometry in which there is a unique line through every pair of distinct points. They also mention that the Euclidean plane version can be proved from the Sylvester-Gallai theorem using induction. The Szemerédi–Trotter theorem A bound on the number of flags determined by a finite set of points and the lines they determine is given by: Theorem (Szemerédi–Trotter): given points and lines in the plane, the number of flags (incident point-line pairs) is: and this bound cannot be improved, except in terms