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.
∠APB = ∠CPD, as a pair of vertical angles.
(*) AP/CP = BP/DP = AB/CD.
The first identity (AP/CP = BP/DP) leads directly to the Intersecting Chords Theorem: AP·DP = BP·CP.
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.
Boat 1 and boat 2, which travel at constant speeds, not necessarily the same, depart at the same time from docks A and C, respectively, on the banks of a circular lake. If they go straight to docks D and B, respectively, they collide. Prove that if boat 1 goes instead straight to dock B and boat 2 goes straight to dock D, they arrive at their destinations simultaneously.
The proof follows (*) by involving the third ratio!
which exactly means that, if boat 1 heads to B and boat 2 heads to D, they arrive there simultaneously.