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Viviani's Theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world. Proof This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA 2010, , p. 96 () Let ABC be an equilateral triangle whose height is ''h'' and whose side is ''a''. Let P be any point inside the triangle, and ''u, s, t'' the distances of P from the sides. Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA. Now, the areas of these triangles are \frac, \frac, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viviani Theorem
Viviani is a surname. Notable people with the surname include: * Anselmo Viviani (born 1915), Italian cross-country skier * Antonio Viviani (1560–1620), painter * Attilio Viviani (born 1996), Italian cyclist * Elia Viviani (born 1989), cyclist * Fabio Viviani (chef) (born 1978) * Fabio Viviani (footballer) (born 1966) * Federico Viviani (footballer born 1981) * Federico Viviani (footballer born 1992) * Guillermo Viviani (1893–1964), Chilean Roman Catholic priest and trade unionist * Giovanni Buonaventura Viviani (1638–1693), composer and violinist * Jody Viviani (born 1982), footballer * Luigi Viviani (other), multiple people * Ottavio Viviani (c. 1579–c. 1641), painter * Raffaele Viviani (1888–1950), author, playwright, and actor * René Viviani (1863–1925), French politician and prime minister * Vincenzo Viviani (1622–1703), mathematician and scientist * Tanios Viviani (born 1961), entrepreneur See also * Square René Viviani * Stadio Alfredo Viviani * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex polygon, convex, star polygon, star or Skew polygon, skew. In the limit (mathematics), limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a Line (geometry), straight line), if the edge length is fixed. General properties ''These properties apply to all regular polygons, whether convex or star polygon, star.'' A regular ''n''-sided polygon has rotational symmetry of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hosted by Wolfram Research, whose stated goal is to bring computational exploration to a large population. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Parents' Choice Award in 2008. Technology The Demonstrations run in '' Mathematica'' 6 or above and in '' Wolfram CDF Player'' which is a free modified version of Wolfram's ''Mathematica'' and available for Windows, Linux and macOS and can operate as a web browser plugin. They typically consist of a very direct user interface to a graphic or visualization, which dynamically recomputes in response to user actions such as moving a slider, clicking a button, or dragging a piece of graphics. Each Demonstration also has a brief description of the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut The Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games. He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website ''Cut-the-Knot'' for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started ''Cut-the-Knot'' in October 1996.Interview with Alexander Bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form , where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. The regular polyhedra There are five convex regular polyhedra, known as the Platonic solids, four regular star polyhedra, the Kepler–Poinsot polyhedra, and five regular compounds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equiangular Polygon
In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. For clarity, a planar equiangular polygon can be called ''direct'' or ''indirect''. A direct equiangular polygon has all angles turning in the same direction in a plane and can include multiple turns. Convex equiangular polygons are always direct. An indirect equiangular polygon can include angles turning right or left in any combination. A skew equiangular polygon may be isogonal, but can't be considered direct since it is nonplanar. A spirolateral ''n''θ is a special case of an ''equiangular polygon'' with a set of ''n'' integer edge lengths repeating sequence until returning to the start, with vertex internal angles θ. Construction An ''equiangular polygon'' can be constructed from a regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a counterexample to the generalization “students are lazy”, and both a counterexample to, and disproof of, the universal quantification “all students are lazy.” In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem. This is most often done by considering a case where a part of the hypothesis is not satisfied and the conclusion of the theorem does not hold. In mathematics In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apothem
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. The word "apothem" can also refer to the length of that line segment and come from the ancient Greek ''ἀπόθεμα'' ("put away, put aside"), made of ''ἀπό'' ("off, away") and ''θέμα'' ("that which is laid down"), indicating a generic line written down. Regular polygons are the only polygons that have apothems. Because of this, all the apothems in a polygon will be congruence (geometry), congruent. For a regular pyramid (geometry), pyramid, which is a pyramid whose base is a regular polygon, the apothem is the slant height of a lateral face; that is, the shortest distance from apex to base on a given face. For a truncated regular pyramid (a regular pyramid with some of its peak removed by a plane (geometry), plane parallel to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Polygon
In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting. Examples All regular polygons and edge-transitive polygons are equilateral. When an equilateral polygon is non-crossing and cyclic (its vertices are on a circle) it must be regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square). A convex equilateral pentagon can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ''ABCD'' add up to 360 degrees of arc, that is :\angle A+\angle B+\angle C+\angle D=360^. This is a special case of the ''n''-gon interior angle sum formula: ''S'' = (''n'' − 2) × 180°. All non-self-crossing quadrilaterals tile the plane, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vincenzo Viviani
Vincenzo Viviani (April 5, 1622 – September 22, 1703) was an Italian mathematician and scientist. He was a pupil of Torricelli and a disciple of Galileo."Viviani" article in the Biography Vincenzo Viviani was born in Florence to the nobles Jacopo di Michelangelo Viviani and Maria Alamanno del Nente. While attending a Jesuit school Viviani studied the . Following the study of humanities, Viviani turned to mathematics. He ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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