Pedal triangle explained

In plane geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle.

More specifically, consider a triangle, and a point that is not one of the vertices . Drop perpendiculars from to the three sides of the triangle (these may need to be produced, i.e., extended). Label the intersections of the lines from with the sides . The pedal triangle is then .

If is not an obtuse triangle and is the orthocenter, then the angles of are, and .^[1]

The location of the chosen point relative to the chosen triangle gives rise to some special cases:

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

The vertices of the pedal triangle of an interior point, as shown in the top diagram, divide the sides of the original triangle in such a way as to satisfy Carnot's theorem:^[2]

$$|AN|^2\; +\; |BL|^2\; +\; |CM|^2\; =\; |NB|^2\; +\; |LC|^2\; +\; |MA|^2.$$

Trilinear coordinates

If has trilinear coordinates, then the vertices of the pedal triangle of are given by

Antipedal triangle

One vertex,, of the antipedal triangle of is the point of intersection of the perpendicular to through and the perpendicular to through . Its other vertices, and, are constructed analogously. Trilinear coordinates are given by

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

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

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

The product of the areas of the pedal triangle of and the antipedal triangle of equals the square of the area of .

Pedal circle

See main article: Pedal circle. 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 not lying on the circumcircle of the triangle, it is known that and its isogonal conjugate have a common pedal circle, whose center is the midpoint of these two points.^[3]

External links

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

Notes and References

1. Web site: Trigonometry/Circles and Triangles/The Pedal Triangle - Wikibooks, open books for an open world. 2020-10-31. en.wikibooks.org.
2. Book: Challenging problems in geometry. limited. Alfred S. Posamentier. Alfred S. Posamentier. Charles T. Salkind. 9780486134864. New York. 829151719. Dover. 1996. 85-86.
3. Book: Honsberger, Ross. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. 1995-01-01. The Mathematical Association of America. 978-0-88385-951-3.