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


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Explanation

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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.


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The theorem bears the name of Philippe de La Hire, a French artist, architect and mathematician born in the 17th 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, OB·OC = R², where R is the radius of the circle. Assume point A lies on polar of point B. Then AC is perpendicular OB. Let D lie on OA with BD perpendicular to the latter. Triangles OAC and OBD are similar (they are both right and share an angle at O.) From here we obtain a proportion OA/OB = OC/OD, which can also be written as

(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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