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Descartes's theorem (plane geometry) Descartes's theorem on total angular defect ; Diaconescu's theorem (mathematical logic) Diller–Dress theorem (field theory) Dilworth's theorem (combinatorics, order theory) Dinostratus' theorem (geometry, analysis) Dimension theorem for vector spaces (vector spaces, linear algebra) Dini's theorem
A 3-coloring of the vertices of a triangulated polygon. The blue vertices form a set of three guards, as few as is guaranteed by the art gallery theorem. However, this set is not optimal: the same polygon can be guarded by only two guards. Steve Fisk's proof is so short and elegant that it was chosen for inclusion in Proofs from THE BOOK. [4]
Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry , which does not assume Euclid's third and ...
The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976. [2] Schröder–Bernstein theorem. In 1896 Schröder published a proof sketch [3] which, however, was shown to be faulty by Alwin Reinhold Korselt in 1911 [4] (confirmed by Schröder). [5] [6] Jordan curve theorem. There has been some controversy about ...
Art Gallery Theorems and Algorithms is a mathematical monograph on topics related to the art gallery problem, on finding positions for guards within a polygonal museum floorplan so that all points of the museum are visible to at least one guard, and on related problems in computational geometry concerning polygons.
The second theorem considers five circles in general position passing through a single point M. Each subset of four circles defines a new point P according to the first theorem. Then these five points all lie on a single circle C. The third theorem considers six circles in general position that pass through a single point M. Each subset of five ...
Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive.
Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity (i.e. is formulable as an elementary theory). As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive ...