Ian McGee's Observation: What is this about?
A Mathematical Droodle


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Copyright © 19962018 Alexander Bogomolny
[Honsberger, p. 33] attributes the following observation to Ian McGee, University of Waterloo:

At points A and B on a circle, equal tangents AP and BQ are drawn as depicted in the applet. Then AB bisects PQ.



Note that, because of the symmetry, the statement is obvious if AB is a diameter of the circle. This is quite surprising that it remains true when A and B are selected on the circle randomly.
Extend AP beyond A to R so that AR = AP. Let BQ meet AP in T, AB meet PQ in S. Now, tangents TA and TB from T to the circle are equal, and since AR = AP = BQ, we also have TR = TQ. Two isosceles triangles ABT and RQT share the same angle at the apex and are, therefore, similar. It follows that QRAB, or QRAS. In ΔPQR, A is the midpoint of side PR and AS is parallel to side QR, it is thus a midline of the triangle. Its other end S is then a midpoint of side PQ.
References
 R. Honsberger, In Pólya's Footsteps, MAA, 1997
Activities
Contact
Front page
Contents
Geometry
Copyright © 19962018 Alexander Bogomolny