Pedal circle information

The pedal circle of the a triangle $ABC$ and a point $P$ in the plane is a special circle determined by those two entities. More specifically for the three perpendiculars through the point $P$ onto the three (extended) triangle sides $a,b,c$ you get three points of intersection $P_{a},P_{b},P_{c}$ 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 $r$ of the pedal circle the following formula holds with $R$ being the radius and $O$ being the center of the circumcircle:^[2]

r={\frac {|PA|\cdot |PB|\cdot |PC|}{2\cdot (R^{2}-|PO|^{2})}}

Note that the denominator in the formula turns 0 if the point $P$ lies on the circumcircle. In this case the three points $P_{a},P_{b},P_{c}$ determine a degenerated circle with an infinite radius, that is a line. This is the Simson line. If $P$ is the incenter of the triangle then the pedal circle is the incircle of the triangle and if $P$ is the orthocenter of the triangle the pedal circle is the nine-point circle.^[3]

If $P$ does not lie on the circumcircle then its isogonal conjugate $Q$ yields the same pedal circle, that is the six points $P_{a},P_{b},P_{c}$ and $Q_{a},Q_{b},Q_{c}$ lie on the same circle. Moreover, the midpoint of the line segment $PQ$ 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.^[4]

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 $ABC$ and the fourth point as the point $P$ , then they define a pedal circle. The four pedal circles you get this way intersect in a common point.^[3]

^ ^a ^b Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, pp. 67–75
^ ^a ^b Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007 (reprint), ISBN 978-0-486-46237-0, pp. 135–144, 155, 240
^ ^a ^b Weisstein, Eric W. "Pedal Circle". MathWorld.
^ Weisstein, Eric W. "Griffiths' Theorem". MathWorld.

[Honsberger-1] Ross Honsberger: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. MAA, 1995, pp. 67–75

[Johnson-2] Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007 (reprint), ISBN 978-0-486-46237-0, pp. 135–144, 155, 240

[mw-3] Weisstein, Eric W. "Pedal Circle". MathWorld.

[4] Weisstein, Eric W. "Griffiths' Theorem". MathWorld.

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