Pedal triangle

File:Pedal Triangle.svg

  Triangle △ABC

  Perpendiculars from point Template:Mvar

  Obtained pedal triangle △LMN

In plane 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 Template:Mvar that is not one of the vertices Template:Mvar. Drop perpendiculars from Template:Mvar to the three sides of the triangle (these may need to be produced, i.e., extended). Label Template:Mvar the intersections of the lines from Template:Mvar with the sides Template:Mvar. The pedal triangle is then △LMN.

If △ABC is not an obtuse triangle and Template:Mvar is the orthocenter, then the angles of △LMN are 180° − 2A, 180° − 2Template:Mvar and 180° − 2C.

The quadrilaterals Template:Mvar are cyclic quadrilaterals.

The location of the chosen point Template:Mvar relative to the chosen triangle △ABC gives rise to some special cases:

• If Template:Mvar is the orthocenter, then △LMN is the orthic triangle.
• If Template:Mvar is the incenter, then △LMN is the intouch triangle.
• If Template:Mvar is the circumcenter, then △LMN is the medial triangle.
• If Template:Mvar is on the circumcircle of the triangle, △LMN collapses to a line (the pedal line or Simson line).

File:Pedal Line.svg

Special case: Template:Mvar is on the circumcircle.

  Triangle △ABC

  Circumcircle of △ABC

  Perpendiculars from Template:Mvar

  Obtained pedal line Template:Mvar

The vertices of the pedal triangle of an interior point Template:Mvar, as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy Carnot's theorem:

$$|AN{|}^2+|BL{|}^2+|CM{|}^2=|NB{|}^2+|LC{|}^2+|MA{|}^2.$$

Trilinear coordinates

If Template:Mvar has trilinear coordinates p : q : r, then the vertices Template:Mvar of the pedal triangle of Template:Mvar are given by $$\begin{array}{ccccccc}L& =& 0& :& q+p\cos C& :& r+p\cos B\\ M& =& p+q\cos C& :& 0& :& r+q\cos A\\ N& =& p+r\cos B& :& q+r\cos A& :& 0\end{array}$$

Antipedal triangle

One vertex, Template:Mvar, of the antipedal triangle of Template:Mvar is the point of intersection of the perpendicular to Template:Mvar through Template:Mvar and the perpendicular to Template:Mvar through Template:Mvar. Its other vertices, Template:Mvar and Template:Mvar, are constructed analogously. Trilinear coordinates are given by $$\begin{array}{ccrcrcr}{L}^{\prime }& =& -(q+p\cos C)(r+p\cos B)& :& (r+p\cos B)(p+q\cos C)& :& (q+p\cos C)(p+r\cos B)\\ M^{\prime }& =& (r+q\cos A)(q+p\cos C)& :& -(r+q\cos A)(p+q\cos C)& :& (p+q\cos C)(q+r\cos A)\\ N^{\prime }& =& (q+r\cos A)(r+p\cos B)& :& (p+r\cos B)(r+q\cos A)& :& -(p+r\cos B)(q+r\cos A)\end{array}$$

For example, the excentral triangle is the antipedal triangle of the incenter.

Suppose that Template:Mvar does not lie on any of the extended sides Template:Mvar, and let P ⁻¹ denote the isogonal conjugate of Template:Mvar. The pedal triangle of Template:Mvar is homothetic to the antipedal triangle of P ⁻¹. The homothetic center (which is a triangle center if and only if Template:Mvar is a triangle center) is the point given in trilinear coordinates by

$$ap(p+q\cos C)(p+r\cos B):bq(q+r\cos A)(q+p\cos C):cr(r+p\cos B)(r+q\cos A)$$

The product of the areas of the pedal triangle of Template:Mvar and the antipedal triangle of P ⁻¹ equals the square of the area of △ABC.

Pedal circle

File:Isogonal points2 pedal circle.svg

The pedal circle of the point Template:Mvar and its isogonal conjugate Template:Mvar are the same.

The pedal circle is defined as the circumcircle of the pedal triangle. Note that the pedal circle is not defined for points lying on the circumcircle of the triangle.

Pedal circle of isogonal conjugates

For any point Template:Mvar not lying on the circumcircle of the triangle, it is known that Template:Mvar and its isogonal conjugate Template:Mvar have a common pedal circle, whose center is the midpoint of these two points.

External links

• Mathworld: Pedal Triangle
• Simson Line
• Pedal Triangle and Isogonal Conjugacy
• pedal triangle and pedal circle - interactive illustration