La Hire's Theorem: What Is It About?
A Mathematical Droodle
Explanation
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Copyright © 1996-2012 Alexander Bogomolny
La Hire's Theorem
The applet suggests the following theorem:
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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:
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If point A lies on the polar of point B, then point B lies on the polar of A.
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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
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Copyright © 1996-2012 Alexander Bogomolny
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