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


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A few words.

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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.


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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

The result is apparently due to M. T. Lemoine, although, both Euler and later Poncelet worked on this statement. A complete proof appears elsewhere. (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

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Copyright © 1996-2017 Alexander Bogomolny

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