Pedal triangle

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".

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 "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"

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".

See also

* Simson line

External links

*
* [http://mathworld.wolfram.com/PedalTriangle.html Mathworld: Pedal Triangle]
* [http://s13a.math.aca.mmu.ac.uk/Geometry/TGeomUnit/TriGeom2.html Java Applet of the Perpendiculars]
* [http://www.cut-the-knot.org/ctk/SimsonLine.shtml#pedalTriangle Simson Line]
* [http://www.cut-the-knot.org/Curriculum/Geometry/OrthologicPedal.shtml Pedal Triangle and Isogonal Conjugacy]


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