The Intersecting Chords Theorem asserts the following very useful fact:

Given a point P in the interior of a circle, pass two lines through P that intersect the circle in points A and D and, respectively, B and C. Then AP·DP = BP·CP.

The proof follows easily from the similarity of triangles ABP and CDP that is a consequence of the equality of their angles:

	BAD = BCD,	as inscribed angles subtended by the same chord BD,
	ABC = ADC,	as inscribed angles subtended by the same chord AC,
	APB = CPD,	as a pair of vertical angles.

Since the proof only uses the equality of the first two ratios, a keen observer might inquire whether the third ratio (AB/CD) is of any use. The following problem (due to N. Vasil'ev) appeared in the Russian Kvant (M26, March-April, 1991, 30) and was further popularized by R. Honsberger in his Mathematical Chestnuts from Around the World.

The proof follows (*) by involving the third ratio!

First, let v₁ and v₂ be the speeds of the boats. The fact that they collide if the first is headed from A to D and the second from C to B, means that they arrive at point P simultaneously, i.e.,

The latter immediately translates into

Taking (*) into account, this implies

or

which exactly means that, if boat 1 heads to B and boat 2 heads to D, they arrive there simultaneously.

References

1. R. Honsberger, Mathematical Chestnuts from Around the World, MAA, 2001, p. 115

Copyright © 1996-2009 Alexander Bogomolny