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Circumgon
In mathematics and particularly in elementary geometry, a circumgon is a geometric figure which circumscribes some circle, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the center of the circle and opposite side on a line that is tangent to the circle. The limiting case in which part or all of the circumgon is a circular arc is permitted. A circumgonal region is the union of those triangular regions. Every triangle is a circumgonal region because it circumscribes the circle known as the incircle of the triangle. Every square is a circumgonal region. In fact, every regular polygon is a circumgonal region, as is more generally every tangential polygon. But not every polygon is a circumgonal region: for example, a non-square rectangle is not. A circumgonal region need not even be a convex polygon: for example, it could consist of three triangular wedges meeting only at the circle's center. All circumgons have comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumgon
In mathematics and particularly in elementary geometry, a circumgon is a geometric figure which circumscribes some circle, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the center of the circle and opposite side on a line that is tangent to the circle. The limiting case in which part or all of the circumgon is a circular arc is permitted. A circumgonal region is the union of those triangular regions. Every triangle is a circumgonal region because it circumscribes the circle known as the incircle of the triangle. Every square is a circumgonal region. In fact, every regular polygon is a circumgonal region, as is more generally every tangential polygon. But not every polygon is a circumgonal region: for example, a non-square rectangle is not. A circumgonal region need not even be a convex polygon: for example, it could consist of three triangular wedges meeting only at the circle's center. All circumgons have comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Polygon
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices. All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi and kites. Characterizations A convex polygon has an incircle if and only if all of its internal angle bisectors are concurrent. This common point is the ''incenter'' (the center of the incircle). There exists a tangential polygon of ''n'' sequential sides ''a''1, ..., ''a''''n'' if and only if the system of equations :x_1+x_2=a_1,\quad x_2+x_3=a_2,\quad \ldots,\quad x_n+x_1=a_n has a solution (''x''1, ..., ''x''''n'') in positive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumscribed Circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an obtuse triangle, the minimum bounding circle has the longest sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circles
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collinearity
In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row". Points on a line In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a Line (geometry), line is typically a Primitive notion, primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A Mathematical model, model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mamikon A
Mamikon A. Mnatsakanian (1942-2021) ( hy, Մամիկոն Մնացականյան) was an Armenian physicist. In 1959, he discovered a new proof of the Pythagorean theorem. He received a Ph.D. in physics in 1969 from Yerevan State University, where he became professor of astrophysics. As an undergraduate he specialized in the development of geometric methods for solving calculus problems by a visual approach that makes no use of formulas, which he later developed into his system of visual calculus. He was a Project Associate at Project Mathematics! at the California Institute of Technology. With co-author Tom Apostol, he won the Mathematical Association of America's award in 2005, 2008, and 2010. When Apostol met Mamikon he wrote, "As a teacher of calculus for more than 50 years and as an author of a couple of textbooks on the subject, I was stunned to learn that many standard problems in calculus can be easily solved by an innovative visual approach that makes no use of formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tom M
Tom or TOM may refer to: * Tom (given name), a diminutive of Thomas or Tomás or an independent Aramaic given name (and a list of people with the name) Characters * Tom Anderson, a character in ''Beavis and Butt-Head'' * Tom Beck, a character in the 1998 American science-fiction disaster movie '' Deep Impact'' * Tom Buchanan, the main antagonist from the 1925 novel ''The Great Gatsby'' * Tom Cat, a character from the ''Tom and Jerry'' cartoons * Tom Lucitor, a character from the American animated series ''Star vs. the Forces of Evil'' * Tom Natsworthy, from the science fantasy novel ''Mortal Engines'' * Tom Nook, a character in ''Animal Crossing'' video game series * Tom Servo, a robot character from the ''Mystery Science Theater 3000'' television series * Tom Sloane, a non-adult character from the animated sitcom ''Daria'' * Talking Tom, the protagonist from the ''Talking Tom & Friends'' franchise * Tom, a character from the '' Deltora Quest'' books by Emily Rodda * Tom, a cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. A strictly convex polygon is a convex polygon such that no line contains two of its edges. In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees. Properties The following properties of a simple polygon are all equivalent to convexity: *Every internal angle is strictly less than 180 degrees. *Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. *The polygon is entirely contained in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a ''square''. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ''ABCD'' would be denoted as . The word rectangle comes from the Latin ''rectangulus'', which is a combination of ''rectus'' (as an adjective, right, proper) and ''angulus'' (angle). A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperboli ... [...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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
