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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 : \begin r^2 & = HA\times HD=HB\times HE=HC\times HF \\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac(a^2+b^2+c^2), \end where ''A, B, C'' denote both the triangle's vertices and the angle measures at those vertices, ''H'' is the orthocenter (the intersection of the triangle's altitudes), ''D'', ''E'', ''F'' are the feet of the altitudes from vertices ''A, B, C'' respectively, ''R'' is the triangle's circumradius (the radius of its circumscribed circle), and ''a'', ''b'', ''c'' are the lengths of the triangle's sides opposite vertices ''A'', ''B'', ''C'' 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 trigonometric formula for the radius shows that the polar c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Circle2
Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * 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 People * Polar (musician), Norwegian electronic music producer Arts, entertainment and media Music Labels and studios * Polar Music, a record label * Polar Studios, music studio of ABBA in Sweden Albums * ''Polar'' (album), second album by the High Water Marks * ''Polars'' (album), an album by the Dutch metal band, Textures Other uses in arts, entertainment and media * ''Polar'' (webcomic), a webcomic and series of graphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
