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Inscribed Square In A Triangle
In elementary geometry, an inscribed square in a triangle is a square whose four vertices all lie on a given triangle. By the pigeonhole principle, two of the square's vertices, and the edge between them, must lie on one of the sides of the triangle. For instance, for the Calabi triangle depicted, the square with horizontal and vertical sides is inscribed; the other two squares in the figure are not inscribed. This is a special case of the inscribed square problem asking for a square whose vertices lie on a simple closed curve. However, although the inscribed square problem remains unsolved in general, it is known to have a solution for every polygon and for every convex set, two special cases that both apply to triangles. Every acute triangle has three inscribed squares, one lying on each of its three sides. In a right triangle there are two inscribed squares, one touching the right angle of the triangle and the other lying on the opposite side. An obtuse triangle has only o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi Triangle
The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an isosceles triangle which is obtuse triangle, obtuse with an irrational number, irrational but algebraic number, algebraic ratio between the lengths of its sides and its base. Definition Consider the largest square that can be placed in an arbitrary triangle. It may be that such a square could be positioned in the triangle in more than one way. If the largest such square can be positioned in three different ways, then the triangle is either an equilateral triangle or the Calabi triangle. Thus, the Calabi triangle may be defined as a triangle that is not equilateral and has three placements for its largest square. Shape The triangle is isosceles which has the same length of sides as . If the ratio of the base to either leg is , we can set that . Then we can consider the following three cases: ;case 1) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Triangle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle is called the '' hypotenuse'' (side c in the figure). The sides adjacent to the right angle are called ''legs'' (or ''catheti'', singular: '' cathetus''). Side a may be identified as the side ''adjacent'' to angle B and ''opposite'' (or ''opposed to'') angle A, while side b is the side adjacent to angle A and opposite angle B. Every right triangle is half of a rectangle which has been divided along its diagonal. When the rectangle is a square, its right-triangular half is isosceles, with two congruent sides and two congruent angles. When the rectangle is not a square, its right-triangular half is scalene. Every triangle whose base is the diameter of a circle and whose apex lies on the circle is a right triangle, with the right angle at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Math Horizons
''Math Horizons'' is a magazine aimed at undergraduates interested in mathematics, published by the Mathematical Association of America. It publishes expository articles about "beautiful mathematics" as well as articles about the culture of mathematics covering mathematical people, institutions, humor, games, cartoons, and book reviews. The MAA gives the Trevor Evans Awards annually to "authors of exceptional articles that are accessible to undergraduates" that are published in ''Math Horizons''. Notes Further reading * External links *''Math Horizons''at JSTOR''Math Horizons'' at Taylor & Francis Online {{mathpublication-stub Mathematics journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. The editor-in-chief is Vadim Ponomarenko (San Diego State University). The journal gives the Lester R. Ford Award annually to "authors of articles of expository excellence" published in the journal. Editors-in-chief The following persons are or have been editor-in-chief: See also *''Mathematics Magazine'' *''Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background. Paid circulation in 2008 was 9,500 and total circulation was 10,000. ''Mathematics Magazine'' is a continuation of ''Mathematics News Letter'' (1926–1934) and ''National Mathematics Magazine'' (1934–1945). Doris Schattschneider became the first female editor of ''Mathematics Magazine'' in 1981. .. The MAA gives the Carl B. Allendoerfer Awards annually "for articles of expository excellence" published in ''Mathematics Magazine''. See also *''American Mathematical Mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Altitude (triangle)
In geometry, an altitude of a triangle is a line segment through a given Vertex (geometry), vertex (called ''apex (geometry), apex'') and perpendicular to a line (geometry), line containing the side or edge (geometry), edge opposite the apex. This (finite) edge and (infinite) line extension are called, respectively, the ''base (geometry), base'' and ''extended side, extended base'' of the altitude. The point (geometry), point at the intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude" or "height", symbol , is the distance between the foot and the apex. The process of drawing the altitude from a vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one-half of the product of an altitude's length and its base's length (symbol ) equals the triangle's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties. The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry. Properties An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obtuse Triangle
An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one '' obtuse angle'' (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles—triangles that are not right triangles because they do not have any right angles (90°). Properties In all triangles, the centroid—the intersection of the medians, each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle. However, while the orthocenter and the circumcenter are in an acute triangle's interior, they are exterior to an obtuse triangle. The orthocenter is the intersection po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acute Triangle
An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one ''obtuse angle'' (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles—triangles that are not right triangles because they do not have any right angles (90°). Properties In all triangles, the centroid—the intersection of the median (geometry), medians, each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle. However, while the orthocenter and the circumcenter are in an acute triangle's interior, they are exterior to an obtuse triangle. The orthocenter is the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary (topology), boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval (mathematics), interval with the property that its epigraph (mathematics), epigraph (the set of points on or above the graph of a function, graph of the function) is a convex set. Convex minimization is a subfield of mathematical optimization, optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
