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Stewart's Theorem
In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746. Statement Let be the lengths of the sides of a triangle. Let be the length of a cevian to the side of length . If the cevian divides the side of length into two segments of length and , with adjacent to and adjacent to , then Stewart's theorem states that :b^2m + c^2n = a(d^2 + mn). A common mnemonic used by students to memorize this equation (after rearranging the terms) is: :\underset = \!\!\!\!\!\! \underset The theorem may be written more symmetrically using signed lengths of segments. That is, take the length to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. In this formulation, the theorem states that if are collinear points, and is any point, then :\left(\overline^2\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
