Pedal triangle

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A triangle ABC in black, the perpendiculars from a point P in blue, and the obtained pedal triangle LMN in red.

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:

The case when P is on the circumcircle, and the pedal triangle degenerates into a line (red).

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 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

The 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 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 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.

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