|
Polar Circle (geometry)
In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is where denote both the triangle's vertex (geometry), vertices and the angle measures at those vertices; is the orthocenter (the intersection of the triangle's altitude (geometry), altitudes); are the feet of the altitudes from vertices respectively; is the triangle's circumradius (the radius of its circumscribed circle); and are the lengths of the triangle's sides opposite vertices respectively.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publications, 2007 (orig. 1960). The first parts of the radius formula reflect the fact that the orthocenter divides the altitudes into segment pairs of equal products. The trigonometry, trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse triangle, obtuse, so one of its angles is obtuse and hence has a negative cosine. Properties A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Circle2
Polar may refer to: Geography * Geographical pole, either of the two points on Earth where its axis of rotation intersects its surface ** Polar climate, the climate common in polar regions ** Polar regions of Earth, locations within the polar circles, referred to as the Arctic and Antarctic Places * Polar, Wisconsin, town in Langlade County, Wisconsin, United States ** Polar (community), Wisconsin, unincorporated community in Langlade County, Wisconsin, United States Arts, entertainment and media * Polar (webcomic), ''Polar'' (webcomic), a webcomic and series of graphic novels by Víctor Santos * Polar (film), ''Polar'' (film), a 2019 Netflix film adaption of the above comic series * ''Polar'', a 2002 novel by T. R. Pearson Music * Polar Music, a record label * Polar Studios, music studio of ABBA in Sweden * Polar (album), ''Polar'' (album), second album by the High Water Marks * Polars (album), ''Polars'' (album), an album by the Dutch metal band, Textures Brands and enterprise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real number, real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as Series (mathematics), infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic function, periodic pheno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Triangle
In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus the incircle of the tangential triangle coincides with the circumcircle of the reference triangle. The circumcenter of the tangential triangle is on the reference triangle's Euler line, as is the center of similitude of the tangential triangle and the orthic triangle (whose vertices are at the feet of the altitude (triangle), altitudes of the reference triangle).Smith, Geoff, and Leversha, Gerry, "Euler and triangle geometry", ''Mathematical Gazette'' 91, November 2007, 436–452. The tangential triangle is homothetic transformation, homothetic to the orthic triangle.Altshiller-Court, Nathan. ''College Geometry'', Dover Publications, 2007 (orig. 1952). A reference triangle and its tangential triangle are in perspective (geometry), persp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nine-point Circle
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of each side of the triangle * The foot of each altitude * The midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet; these line segments lie on their respective altitudes). The nine-point circle is also known as Feuerbach's circle (after Karl Wilhelm Feuerbach), Euler's circle (after Leonhard Euler), Terquem's circle (after Olry Terquem), the six-points circle, the twelve-points circle, the -point circle, the medioscribed circle, the mid circle or the circum-midcircle. Its center is the nine-point center of the triangle. Nine Significant Points of Nine Point Circle The diagram above shows the nine significant points of the nine-point circle. Points are the midpoints of the thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pole And Polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole. Properties Pole and polar have several useful properties: * If a point P lies on the line ''l'', then the pole L of the line ''l'' lies on the polar ''p'' of point P. (La Hire's theorem) * If a point P moves along a line ''l'', its polar ''p'' rotates about the pole L of the line ''l''. * If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points. * If a point lies on the conic section, its polar is the tangent through this point to the conic section. * If a point P lies on its own polar line, then P is on the conic section. * Each line has, with respect to a non-degenerated conic section, exactly one pole. Special case o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coaxal Circles
In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned ancient Greek geometer. Definition The Apollonian circles are defined in two different ways by a line segment denoted . Each circle in the first family (the blue circles in the figure) is associated with a positive real number , and is defined as the locus of points such that the ratio of distances from to and to equals , \left\. For values of close to zero, the corresponding circle is close to , while for values of close to , the corresponding circle is close to ; for the intermediate value , the circle degenerates to a line, the perpendicular bisector of . The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Quadrilateral
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a ''complete quadrilateral'' is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used. The complete quadrilateral has also been called a Pasch configuration, especially in the context of Steiner triple systems. Diagonals The six lines of a complete quadrangle meet in pairs to form three additional points called the ''diagonal points'' of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically used for lines and planes that intersect to form a right angle, whereas ''orthogonal'' is used in generalizations, such as ''orthogonal vectors'' or ''orthogonal curves''. ''Orthogonality'' is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics Optics In optics, polarization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocentric System
In geometry, an orthocentric system is a set (mathematics), set of four point (geometry), points on a plane (mathematics), plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and the four circles passing through any three of the four points have the same radius. If four points form an orthocentric system, then ''each'' of the four points is the orthocenter of the other three. These four possible triangles will all have the same nine-point circle. Consequently these four possible triangles must all have circumcircles with the same circumradius. The common nine-point circle The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Circle4
Polar may refer to: Geography * Geographical pole, either of the two points on Earth where its axis of rotation intersects its surface ** Polar climate, the climate common in polar regions ** Polar regions of Earth, locations within the polar circles, referred to as the Arctic and Antarctic Places * Polar, Wisconsin, town in Langlade County, Wisconsin, United States ** Polar (community), Wisconsin, unincorporated community in Langlade County, Wisconsin, United States Arts, entertainment and media * ''Polar'' (webcomic), a webcomic and series of graphic novels by Víctor Santos * ''Polar'' (film), a 2019 Netflix film adaption of the above comic series * ''Polar'', a 2002 novel by T. R. Pearson Music * Polar Music, a record label * Polar Studios, music studio of ABBA in Sweden * ''Polar'' (album), second album by the High Water Marks * ''Polars'' (album), an album by the Dutch metal band, Textures Brands and enterprises * Polar Air Cargo, an American airline * Polar Airl ... [...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] |
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] |
