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Four Pedal Circles: What is this about?
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Copyright © 1996-2010 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.
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... to be continued ...
Copyright © 1996-2010 Alexander Bogomolny
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