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- Pedal_triangle abstract "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 L that is one of the vertices A, B, C. Drop medians 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.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 is on the circumcircle of the triangle, LMN collapses to a line. This is then called the pedal line, or sometimes the Simson line after Robert Simson.If P has trilinear coordinates p : q : r, then the vertices L,M,N of the pedal triangle of Cake are given byL = 0 : q + p cos C : r + p cos BM = p + q cos C : 0 : r + q cos AN = p + r cos B : q + r cos A : 0The A-vertex, L', of the antipedal triangle of P is the point of intersection of the perpendicular to BP through B and the perpendicular to CP through C. The B-vertex, M ', and the C-vertex, N ', are constructed analogously. Trilinear coordinates are given byL' = - (q + p cos C)(r + p cos B) : (r + p cos B)(p + q cos C) : (q + p cos C)(p + r cos B)M' = (r + q cos A)(q + p cos C) : - (r + q cos A)(p + q cos C) : (p + q cos C)(q + r cos A)N' = (q + r cos A)(r + p cos B) : (p + r cos B)(r + q cos A) : - (p + r cos B)(q + r cos A)For example, the excentral triangle is the antipedal triangle of the incenter.Suppose that P does not lie on a sideline, BC, CA, AB, and let P - 1 denote the isogonal conjugate of P. The pedal triangle of P is homothetic to the antipedal triangle of P - 1. The homothetic center (which is a triangle center if and only if P is a triangle center) is the point given in trilinear coordinates by ap(p + q cos C)(p + r cos B) : bq(q + r cos A)(q + p cos C) : cr(r + p cos B)(r + q cos A).Another theorem about the pedal triangle of P and the antipedal triangle of P - 1 is that the product of their areas equals the square of the area of triangle ABC. The point from which perpendiculars are drawn should be orthocentre then and only then it will be called as pedal triangle.".
- Pedal_triangle thumbnail Pedal_Triangle.svg?width=300.
- Pedal_triangle wikiPageExternalLink PedalTriangle.html.
- Pedal_triangle wikiPageExternalLink TriGeom2.html.
- Pedal_triangle wikiPageExternalLink OrthologicPedal.shtml.
- Pedal_triangle wikiPageExternalLink pedalTriangle.
- Pedal_triangle wikiPageID "152983".
- Pedal_triangle wikiPageRevisionID "582759709".
- Pedal_triangle hasPhotoCollection Pedal_triangle.
- Pedal_triangle id "2277".
- Pedal_triangle title "Simson's line".
- Pedal_triangle subject Category:Triangle_geometry.
- Pedal_triangle comment "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 L that is one of the vertices A, B, C. Drop medians 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.".
- Pedal_triangle label "Fußpunktdreieck".
- Pedal_triangle label "Pedal triangle".
- Pedal_triangle label "Triangolo pedale".
- Pedal_triangle label "Voetpuntsdriehoek".
- Pedal_triangle label "Подерный треугольник".
- Pedal_triangle label "مثلث دواسة".
- Pedal_triangle sameAs Fußpunktdreieck.
- Pedal_triangle sameAs Triangolo_pedale.
- Pedal_triangle sameAs Voetpuntsdriehoek.
- Pedal_triangle sameAs m.013yzv.
- Pedal_triangle sameAs Q478728.
- Pedal_triangle sameAs Q478728.
- Pedal_triangle wasDerivedFrom Pedal_triangle?oldid=582759709.
- Pedal_triangle depiction Pedal_Triangle.svg.
- Pedal_triangle isPrimaryTopicOf Pedal_triangle.
