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

A Mathematical Droodle

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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 17^{th} century.

### Proof

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
- 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

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Copyright © 1996-2018 Alexander Bogomolny65979676