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Pedal Triangle
In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the three sides of the triangle (these may need to be produced, i.e., extended). Label ''L'', ''M'', ''N'' the intersections of the lines from ''P'' with the sides ''BC'', ''AC'', ''AB''. The pedal triangle is then ''LMN''. If ABC is not an obtuse triangle, P is the orthocenter then the angles of LMN are 180°−2A, 180°−2B and 180°−2C. The location of the chosen point ''P'' relative to the chosen triangle ''ABC'' gives rise to some special cases: * If ''P = '' orthocenter, then ''LMN = '' orthic triangle. * If ''P = ''incenter, then ''LMN = ''intouch triangle. * If ''P = '' circumcenter, then ''LMN = '' medial triangle. If ''P'' is on the circumcircle of the triangle, ''LMN'' collapses to a line. This is then called the pedal line, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simson Line
In geometry, given a triangle and a point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first published, however, by William Wallace in 1799. The converse is also true; if the three closest points to on three lines are collinear, and no two of the lines are parallel, then lies on the circumcircle of the triangle formed by the three lines. Or in other words, the Simson line of a triangle and a point is just the pedal triangle of and that has degenerated into a straight line and this condition constrains the locus of to trace the circumcircle of triangle . Equation Placing the triangle in the complex plane, let the triangle with unit circumcircle have vertices whose locations have complex coordinates , , , and let P with complex coordinates be a point on the circumcircle. The Simson line is the set of points satisfyingTodor Zaharinov, " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homothetic Transformation
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by the rule : \overrightarrow=k\overrightarrow for a fixed number k\ne 0. Using position vectors: :\mathbf x'=\mathbf s + k(\mathbf x -\mathbf s). In case of S=O (Origin): :\mathbf x'=k\mathbf x, which is a uniform scaling and shows the meaning of special choices for k: :for k=1 one gets the ''identity'' mapping, :for k=-1 one gets the ''reflection'' at the center, For 1/k one gets the ''inverse'' mapping defined by k. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if k>0) or reverse (if k<0) the direction of all vectors. Together with the translations, all homotheties of an affi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isogonal Conjugate
__notoc__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (This definition applies only to points not on a sideline of triangle .) This is a direct result of the trigonometric form of Ceva's theorem. The isogonal conjugate of a point is sometimes denoted by . The isogonal conjugate of is . The isogonal conjugate of the incentre is itself. The isogonal conjugate of the orthocentre is the circumcentre . The isogonal conjugate of the centroid is (by definition) the symmedian point . The isogonal conjugates of the Fermat points are the isodynamic points and vice versa. The Brocard points are isogonal conjugates of each other. In trilinear coordinates, if X=x:y:z is a point not on a sideline of triangle , then its isogonal conjugate is \tfrac : \tfrac : \tfrac. For this reason, the isogona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excentral Triangle
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trilinear Coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carnot's Theorem (perpendiculars)
Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection. The theorem can also be thought of as a generalization of the Pythagorean theorem. Theorem For a triangle \triangle ABC with sides a, b, c consider three lines that are perpendicular to the triangle sides and intersect in a common point F. If P_a, P_b, P_c are the pedal points of those three perpendiculars on the sides a, b, c, then the following equation holds: : , AP_c, ^2+, BP_a, ^2+, CP_b, ^2=, BP_c, ^2+, CP_a, ^2+, AP_b, ^2 The converse of the statement above is true as well, that is if the equation holds for the pedal points of three perpendiculars on the three triangle sides then they intersect in a common point. Therefore, the equation provides a necessary and sufficient condition. Special cases If the triangle \triangle ABC has a right angle in C, then we can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Simson
Robert Simson (14 October 1687 – 1 October 1768) was a Scottish mathematician and professor of mathematics at the University of Glasgow. The Simson line is named after him.Robert Simson University of Glasgow (multi-tab page) Life The eldest son of John Simson of Kirktonhall, in , Robert Simson was intended for the Church, but the bent of his mind was towards mathematics. He was educated at the University of Glasgow and graduated M.A. When the prospect opened of his succeeding to the[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcircle
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 longes ... [...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 geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pedal Line
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control playback of voice dictations Geometry * Pedal curve, a curve derived by construction from a given curve * Pedal triangle, a triangle obtained by projecting a point onto the sides of a triangle Music Albums * ''Pedals'' (Rival Schools album) * ''Pedals'' (Speak album) Other music * Bass drum pedal, a pedal used to play a bass drum while leaving the drummer's hands free to play other drums with drum sticks, hands, etc. * Effects pedal, a pedal used commonly for electric guitars * Pedal keyboard, a musical keyboard operated by the player's feet * Pedal harp, a modern orchestral harp with pedals used to change the tuning of its strings * Pedal point, a type of nonchord tone, usually in the bass * Pedal tone, a fundamental tone played ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
