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

A Mathematical Droodle

What if applet does not run? |

|Activities| |Contact| |Front page| |Contents| |Geometry| |Eye opener|

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. |

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

- D. Wells,
*Curious and Interesting Geometry*, Penguin Books, 1991

## Poles and Polars

- Poles and Polars
- Brianchon's Theorem
- Complete Quadrilateral
- Harmonic Ratio
- Harmonic Ratio in Complex Domain
- Inversion
- Joachimsthal's Notations
- La Hire's Theorem
- La Hire's Theorem, a Variant
- La Hire's Theorem in Ellipse
- Nobbs' Points, Gergonne Line
- Polar Circle
- Pole and Polar with Respect to a Triangle
- Poles, Polars and Quadrilaterals
- Straight Edge Only Construction of Polar
- Tangents and Diagonals in Cyclic Quadrilateral
- Secant, Tangents and Orthogonality
- Poles, Polars and Orthogonal Circles
- Seven Problems in Equilateral Triangle, Solution to Problem 1

|Activities| |Contact| |Front page| |Contents| |Geometry| |Eye opener|

Copyright © 1996-2018 Alexander Bogomolny

68949825