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Petr–Douglas–Neumann Theorem
In geometry, the Petr–Douglas–Neumann theorem (or the PDN-theorem) is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908. The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940 and also by B H Neumann (1909–2002) in 1941. The naming of the theorem as ''Petr–Douglas–Neumann theorem'', or as the ''PDN-theorem'' for short, is due to Stephen B Gray. This theorem has also been called Douglas's theorem, the Douglas–Neumann theorem, the Napoleon–Douglas–Neumann theorem and Petr's theorem. The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals. Statement of the theorem The Pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral with successiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sqr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PDN Theorem For Pentagon
__NOTOC__ PDN may refer to: Digital * PDN Gateway (Packet Data Network Gateway), * PDN Mail, electronic messaging developed by Prime Computer * Public data network * Paint.NET File extension * Portable Draughts Notation (.PDN), the standard computer format for recording draughts games * .PDN, the native format for the graphics editing program Paint.NET Publication * '' Pacific Daily News'', a newspaper based on Guam * '' Photo District News'', US publication for photographers Entity * Department of Physiology, Development and Neuroscience, University of Cambridge * National Democratic Party (Argentina) The National Democratic Party ( es, Partido Demócrata Nacional, PDN) was a conservative political party in Argentina created in 1931. It was generally known simply as Conservative Party ( es, Partido Conservador). Along with the ''Antipersonal ... () Other * ISO 639:pdn or Podena language, spoken in Indonesia * Phenotypic disease network (PDN) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PDN Theorem For Quadrilateral As Van Aubels Theorem
__NOTOC__ PDN may refer to: Digital * PDN Gateway (Packet Data Network Gateway), * PDN Mail, electronic messaging developed by Prime Computer * Public data network * Paint.NET File extension * Portable Draughts Notation (.PDN), the standard computer format for recording draughts games * .PDN, the native format for the graphics editing program Paint.NET Publication * '' Pacific Daily News'', a newspaper based on Guam * '' Photo District News'', US publication for photographers Entity * Department of Physiology, Development and Neuroscience, University of Cambridge * National Democratic Party (Argentina) The National Democratic Party ( es, Partido Demócrata Nacional, PDN) was a conservative political party in Argentina created in 1931. It was generally known simply as Conservative Party ( es, Partido Conservador). Along with the ''Antipersonal ... () Other * ISO 639:pdn or Podena language, spoken in Indonesia * Phenotypic disease network (PDN) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PDN Theorem For Self Intersection Quadrilateral Case2
__NOTOC__ PDN may refer to: Digital * PDN Gateway (Packet Data Network Gateway), * PDN Mail, electronic messaging developed by Prime Computer * Public data network * Paint.NET File extension * Portable Draughts Notation (.PDN), the standard computer format for recording draughts games * .PDN, the native format for the graphics editing program Paint.NET Publication * '' Pacific Daily News'', a newspaper based on Guam * '' Photo District News'', US publication for photographers Entity * Department of Physiology, Development and Neuroscience, University of Cambridge * National Democratic Party (Argentina) The National Democratic Party ( es, Partido Demócrata Nacional, PDN) was a conservative political party in Argentina created in 1931. It was generally known simply as Conservative Party ( es, Partido Conservador). Along with the ''Antipersonal ... () Other * ISO 639:pdn or Podena language, spoken in Indonesia * Phenotypic disease network (PDN) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PDN Theorem For Self Intersection Quadrilateral Case1
__NOTOC__ PDN may refer to: Digital * PDN Gateway (Packet Data Network Gateway), * PDN Mail, electronic messaging developed by Prime Computer * Public data network * Paint.NET File extension * Portable Draughts Notation (.PDN), the standard computer format for recording draughts games * .PDN, the native format for the graphics editing program Paint.NET Publication * '' Pacific Daily News'', a newspaper based on Guam * '' Photo District News'', US publication for photographers Entity * Department of Physiology, Development and Neuroscience, University of Cambridge * National Democratic Party (Argentina) The National Democratic Party ( es, Partido Demócrata Nacional, PDN) was a conservative political party in Argentina created in 1931. It was generally known simply as Conservative Party ( es, Partido Conservador). Along with the ''Antipersonal ... () Other * ISO 639:pdn or Podena language, spoken in Indonesia * Phenotypic disease network (PDN) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can be defined between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. Definitions A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek διαγώνιος ''diagonios'', "from angle to angle" (from διά- ''dia-'', "through", "across" and γωνία ''gonia'', "angle", related to ''gony'' "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as ''diagonus'' ("slanting line"). In matrix algebra, the diagonal of a square matrix consists of the entries on the line from the top left corner to the bottom right corner. There are also other, non-mathematical uses. Non-mathematical uses In engineering, a diagonal brace is a beam used to brace a rectangular structure (such as scaffolding) to withstand strong forces pushing into it; although called a diagonal, due to practical consideration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point (geometry), point where two or more curves, line (geometry), lines, or edge (geometry), edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedron, polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a vertex is called "convex set, convex" if the internal an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
