La Hire's Theorem: What Is It About?
A Mathematical Droodle
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Copyright © 19962018 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

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Copyright © 19962018 Alexander Bogomolny