La Hire's Theorem: What Is It About?
A Mathematical Droodle
| What if applet does not run? |
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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. |
| 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
72199769