La Hire's Theorem: What Is It About?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2018 Alexander Bogomolny

La Hire's Theorem

The applet suggests the following theorem:

Let there be two points A and B outside the circle O. From points A and B draw tangents AC, AD, BE, BF to the circle. Then if B lies on CD, then A lies on EF.


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What if applet does not run?

The theorem is a particular case of a more general La Hire's theorem:

If point A lies on the polar of point B, then point B lies on the polar of A.

Indeed, for a point P outside the circle of reference, the polar passes through the tangency points of the tangents drawn from P to the circle.

References

  1. D. Wells, Curious and Interesting Geometry, Penguin Books, 1991

Poles and Polars

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Copyright © 1996-2018 Alexander Bogomolny

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