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Trapezoids
In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are called the ''legs'' or ''lateral sides''. (If the trapezoid is a parallelogram, then the choice of bases and legs is arbitrary.) A trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If ''ABCD'' is a convex trapezoid, then ''ABDC'' is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids. Definitions ''Trapezoid'' can be defined exclusively or inclusively. Under an exclusive definition a trapezoid is a quadrilateral having pair of parallel sides, with the other pair of opposite sides non-parallel. Parallelograms including rhombi, rectangles, and squares are then not considered to be trapezoids. Under an inclusive definition, a trapezoid is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Trapezoid
In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure, or as a trapezoid whose diagonals have equal length. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram), and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base). Special cases Trapezoid is defined as a quadrilateral having exactly one pair of parallel sides, with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), 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 polygon, simple (not self-intersecting), or complex polygon, complex (self-intersecting, or crossed). Simple quadrilaterals are either convex polygon, convex or concave polygon, concave. The Internal and external angle, interior angles of a simple (and Plane (geometry), planar) quadrilateral ''ABCD'' add up to 360 Degree (angle), degrees, that is :\angle A+\angle B+\angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossed 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, 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° (here, n=4). All non-self-crossing quadrilaterals tile the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posidonius
Posidonius (; , "of Poseidon") "of Apameia" (ὁ Ἀπαμεύς) or "of Rhodes" (ὁ Ῥόδιος) (), was a Greeks, Greek politician, astronomer, astrologer, geographer, historian, mathematician, and teacher native to Apamea (Syria), Apamea, Syria. He was considered the most learned man of his time and, possibly, of the entire Stoicism, Stoic school. After a period learning Stoicism, Stoic philosophy from Panaetius in Athens, he spent many years in travel and scientific researches in Spain, Africa, Italy, Gaul, Liguria, Sicily and on the eastern shores of the Adriatic. He settled as a teacher at Rhodes where his fame attracted numerous scholars. Next to Panaetius he did most, by writings and personal lectures, to spread Stoicism to the Roman world, and he became well known to many leading men, including Pompey and Cicero. His works are now lost, but they proved a mine of information to later writers. The titles and subjects of more than twenty of them are known. In common w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides , and the hypotenuse , sometimes called the Pythagorean equation: :a^2 + b^2 = c^2 . The theorem is named for the Ancient Greece, Greek philosopher Pythagoras, born around 570 BC. The theorem has been Mathematical proof, proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both Geometry, geometric proofs and Algebra, algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Garfield's Proof Of The Pythagorean Theorem
Garfield's proof of the Pythagorean theorem is an original proof of the Pythagorean theorem discovered by James A. Garfield (November 19, 1831 – September 19, 1881), the 20th president of the United States. The proof appeared in print in the ''Journal of Education, New-England Journal of Education'' (Vol. 3, No.14, April 1, 1876). At the time of the publication of the proof Garfield was a Congressman from Ohio. He assumed the office of President on March 4, 1881, and served in that position only for a brief period up to September 19, 1881 when he died following his assassination in July. Garfield was the only President of the United States to have contributed anything original to mathematics. The proof is nontrivial and, according to the historian of mathematics William Dunham (mathematician), William Dunham, "Garfield's is really a very clever proof." The proof appears as the 231st proof in ''The Pythagorean Proposition'', a compendium of 370 different proofs of the Pythagorean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James A
James may refer to: People * James (given name) * James (surname) * James (musician), aka Faruq Mahfuz Anam James, (born 1964), Bollywood musician * James, brother of Jesus * King James (other), various kings named James * Prince James (other) * Saint James (other) Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, York, James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Film and television * James (2005 film), ''James'' (2005 film), a Bollywood film * James (2008 film), ''James'' (2008 film), an Irish short film * James (2022 film), ''James'' (2022 film), an Indian Kannada-language film * "James", a television Adventure Time (season 5)#ep42, ... [...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] |
Right Angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin ''angulus rectus''; here ''rectus'' means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to Euclidean vector, vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry. Etymology The meaning of ''right'' in ''right angle'' possibly refers to the Classical Latin, Latin adjective ''rectus'' 'erect, straight, upright, perp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiparallelogram
In geometry, an antiparallelogram is a type of list of self-intersecting polygons, self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel (geometry), parallel. Instead, each pair of sides is antiparallel lines, antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also called contraparallelograms or crossed parallelograms. Antiparallelograms occur as the vertex figures of certain nonconvex uniform polyhedron, nonconvex uniform polyhedra. In the theory of four-bar linkages, the linkages with the form of an antiparallelogram are also called butterfly linkages or bow-tie linkages, and are used in the design of non-circular gears. In celestial mechan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a Bounded set, bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its projective duality, dual problem of intersecting Half-space (geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
