# A Median in Touching Circles: What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet attempts to illustrate a problem once suggested for but not used at an international math olympiad ([Honsberger, p. 29]):

With a reference to the applet, two circles with diameters AB and AE are tangent internally at A. E trisects AB. O is the center of the big circle. P is a point on the small circle and AP crosses the big one in C. BC and OP intersect in D. Prove that

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The two circles are homothetic at the point of tangency and the coefficient 3/2 (from the fact that E trisects AB.) Under this homothety, point C corresponds to P so that AC = 3/2·AP. In other words, P trisects AC.

On the other hand, O being the center of the big circle,

(Nathan Bowler has observed that the result follows an application of Menelaus' theorem to ΔABC.)

### References

- R. Honsberger,
*In Pólya's Footsteps*, MAA, 1997

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Copyright © 1996-2018 Alexander Bogomolny