Four Pedal Circles: What is this about?
A Mathematical Droodle


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

A few words.

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Copyright © 1996-2017 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.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

... to be continued ...

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

Copyright © 1996-2017 Alexander Bogomolny

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