# Four Pedal Circles: What is this about?

A Mathematical Droodle

What if applet does not run? |

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Copyright © 1996-2018 Alexander Bogomolny

A complete quadrangle is a configuration of 4 points and 6 lines. Taken by three, the points define 4 triangles. Any point P in the plane of the quadrangle relates to each of those a pedal triangle and an associated pedal circle. The applet purports to suggest the following statement:

The four pedal circles defined by a point and a complete quadrangle are concurrent. |

There are a few related facts.

If the quadrilateral ABCD is cyclic, the four 9-point circles and the simsons of each of the points with respect to the triangle formed by the other three all meet in a point.

A generalization of the above concyclicity: for an arbitrary quadrilateral ABCD, the 9-point circles of triangles BCD, CDA, DAB, ABC and the pedal circles of the corresponding remaining point, are concurrent.

What if applet does not run? |

... to be continued ...

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny

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