# Four 9-Point Circles in a Quadrilateral

What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 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.

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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

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

### Nine Point Circle

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Copyright © 1996-2018 Alexander Bogomolny