Pedal circle explained
The pedal circle of the a triangle
and a point
in the plane is a special circle determined by those two entities. More specifically for the three perpendiculars through the point
onto the three (extended) triangle sides
you get three points of intersection
and the circle defined by those three points is the pedal circle. By definition the pedal circle is the
circumcircle of the
pedal triangle.
[1] [2] For radius
of the pedal circle the following formula holds with
being the radius and
being the center of the circumcircle:
[2] | r= | |PA| ⋅ |PB| ⋅ |PC| |
| 2 ⋅ (R2-|PO|2) |
Note that the denominator in the formula turns 0 if the point
lies on the circumcircle. In this case the three points
determine a degenerated circle with an infinite radius, that is a line. This is the
Simson line. If
is the
incenter of the triangle then the pedal circle is the
incircle of the triangle and if
is the orthocenter of the triangle the pedal circle is the
nine-point circle.
If
does not lie on the circumcircle then its
isogonal conjugate
yields the same pedal circle, that is the six points
and
lie on the same circle. Moreover, the
midpoint of the line segment
is the center of that pedal circle.
[1] Griffiths' theorem states that all the pedal circles for a points located on a line through the center of the triangle's circumcircle share a common (fixed) point.
Consider four points with no three of them being on a common line. Then you can build four different subsets of three points. Take the points of such a subset as the vertices of a triangle
and the fourth point as the point
, then they define a pedal circle. The four pedal circles you get this way intersect in a common point.
References
- [Ross Honsberger]
- Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007 (reprint), ISBN 978-0-486-46237-0, pp. 135–144, 155, 240
External links
