Four 9-Point Circles in a Quadrilateral: What is this about?
A Mathematical Droodle

(Click repeatedly click on the applet. You will obtain some additional information that might help you surmise what it is about.)

A few words.

Copyright © 1996-2009 Alexander Bogomolny

Any quadrilateral ABCD defines four triangles: BCD, CDA, DAB, ABC. In each of the triangles one may consider associate remarkable points and lines. The above applet is specifically concerned with their 9-point circles of those triangles. Elsewhere we established the following result:

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.

Presently, we present a generalization of this claim, by removing the requirement of cyclicity: 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.

The result is apparently due to M. T. Lemoine, see also [Coolidge, p. 123].

References

1. J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971

Nine Point Circle

• Feuerbach's Theorem
• Feuerbach's Theorem: a Proof
• Four 9-Point Circles in a Quadrilateral
• Four Triangles, One Circle
• Hart Circle
• Incidence in Feuerbach's Theorem
• Six Point Circle
• Nine Point Circle
• 6 to 9 Point Circle
  • Six Concyclic Points
• Six Concyclic Points II
• Bevan's Point and Theorem

Copyright © 1996-2009 Alexander Bogomolny