## Concentric Circles

The region between two Concentric circles of different Radii is called an Annulus.

Given two concentric circles with Radii and , what is the probability that a chord chosen at random from the outer circle will cut across the inner circle? Depending on how the random'' Chord is chosen, 1/2, 1/3, or 1/4 could all be correct answers.

1. Picking any two points on the outer circle and connecting them gives 1/3.

2. Picking any random point on a diagonal and then picking the Chord that perpendicularly bisects it gives 1/2.

3. Picking any point on the large circle, drawing a line to the center, and then drawing the perpendicularly bisected Chord gives 1/4.
So some care is obviously needed in specifying what is meant by random'' in this problem.

Given an arbitrary Chord to the larger of two concentric Circles centered on , the distance between inner and outer intersections is equal on both sides . To prove this, take the Perpendicular to passing through and crossing at . By symmetry, it must be true that and are equal. Similarly, and must be equal. Therefore, equals . Incidentally, this is also true for Homeoids, but the proof is nontrivial.