La Hire's Theorem: What Is It About?
A Mathematical Droodle
Copyright © 1996-2018 Alexander Bogomolny
La Hire's Theorem
The applet suggests the following theorem:
If point A lies on the polar of point B, then point B lies on the polar of A.
The theorem bears the name of Philippe de La Hire, a French artist, architect and mathematician born in the 17th century.
By definition, the polar of B is perpendicular to OB, where O is the center of the circle of reference at the inverse image C of B. Therefore,
|(1)||OA·OD = OB·OC = R².|
Thus also OA·OD = R², so that D is the inverse image of A and BD is its polar, which proves the theorem.
Note that the proof works in all cases, except where the three points O, A, and B are collinear. The polars of A and B are then parallel, the triangles OAC and OBD degenerate into straight line segments, and the proof fails. But then obviously A coincides with C, and (1) still holds. In this case, A and C are just inverse images of each other.
Poles and Polars
- Poles and Polars
- Brianchon's Theorem
- Complete Quadrilateral
- Harmonic Ratio
- Harmonic Ratio in Complex Domain
- 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