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Concept
Pedal triangle
Summary
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, or sometimes the Simson line after Robert Simson.
The vertices of the pedal triangle of an interior point P, as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy Carnot's theorem:
If P has trilinear coordinates p : q : r, then the vertices L,M,N of the pedal triangle of P are given by
L = 0 : q + p cos C : r + p cos B
M = p + q cos C : 0 : r + q cos A
N = p + r cos B : q + r cos A : 0
One 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. Its other vertices, M ' and N ', are constructed analogously. Trilinear coordinates are given by
L = − (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 any of the extended sides BC, CA, AB, and let P−1 denote the isogonal conjugate of P.
Official source
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