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

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Explanation

### 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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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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