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

Explanation

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[Honsberger, p. 33] attributes the following observation to Ian McGee, University of Waterloo:

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 QR||AB, or QR||AS. 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

1. R. Honsberger, In Pólya's Footsteps, MAA, 1997

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