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Pedal Triangle
In plane geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle , and a point that is not one of the vertices . Drop perpendiculars from to the three sides of the triangle (these may need to be produced, i.e., extended). Label the intersections of the lines from with the sides . The pedal triangle is then . If is not an obtuse triangle and is the orthocenter, then the angles of are , and . The quadrilaterals are cyclic quadrilaterals. The location of the chosen point relative to the chosen triangle gives rise to some special cases: * If is the orthocenter, then is the orthic triangle. * If is the incenter, then is the intouch triangle. * If is the circumcenter, then is the medial triangle. *If is on the circumcircle of the triangle, collapses to a line (the ''pedal line'' or ''Simson line''). The vertices of the pedal triangle of an interior point , as shown in the top diag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Circumcenter
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case , a cyclic quadrilateral. All rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is. Straightedge and compass construction The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors. For three non-collinear points, these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector is equidistant from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Homothetic Transformation
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a Transformation (mathematics), transformation of an affine space determined by a point called its ''center'' and a nonzero number called its ''ratio'', which sends point to a point 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 Similarity (geometry), similarities that fix a point and either preserve (if k>0) or reverse (if k<0) the direction of all vectors. Together with the Translation (geometry), translations, all homotheties of an affine (or Euclidean) space form a group (mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 isogonal co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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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 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 t .... 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 peda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Simson Line
In geometry, given a triangle and a Point (geometry), 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 (mathematician), William Wallace in 1799, and is sometimes called the Wallace line. The Theorem#Converse, 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 (mathematics), 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 l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Circumcircle
In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection (geometry), intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case , a cyclic quadrilateral. All rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is. Straightedge and compass construction The circumcenter of a triangle can be Compass-and-straightedge construction, constructed by drawing any two of the three Bisection#Perpendicular bisectors, perpendicular bisectors. For three non-collinear points, these two lines cannot be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Medial Triangle
In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is not the same thing as the median triangle, which is the triangle whose sides have the same lengths as the medians of . Each side of the medial triangle is called a ''midsegment'' (or ''midline''). In general, a midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to half the length of the third side. Properties The medial triangle can also be viewed as the image of triangle transformed by a homothety centered at the centroid with ratio -1/2. Thus, the sides of the medial triangle are half and parallel to the corresponding sides of triangle ABC. Hence, the medial triangle is inversely similar and shares the same centroid and medians w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |