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Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): [16] "Let the following be postulated": "To draw a straight line from any point to any point." "To produce [extend] a finite straight line continuously in a straight line."
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.
Problems and Theorems in Analysis (German: Aufgaben und Lehrsätze aus der Analysis) is a two-volume problem book in analysis by George Pólya and Gábor Szegő. Published in 1925, the two volumes are titled (I) Series. Integral Calculus. Theory of Functions.; and (II) Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry.
[4] Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): [5] Let the following be postulated: To draw a straight line from any point to any point. To produce (extend) a finite straight line continuously in a straight line.
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]
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 ...
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
This, for instance, applies to all theorems in Euclid's Elements, Book I. An example of a theorem of Euclidean geometry which cannot be so formulated is the Archimedean property: to any two positive-length line segments S 1 and S 2 there exists a natural number n such that nS 1 is longer than S 2.