Problem:
In a circle, parallel chords of lengths 2,3 and 4 determine central angles of Ξ±,Ξ² and Ξ±+Ξ² radians, respectively, where Ξ±+Ξ²<Ο. If cosΞ±, which is a positive rational number, is expressed as a fraction in lowest terms, what is the sum of its numerator and denominator?
Solution:
Since any two chords of equal length subtend equal angles, the parallelism of the chords is irrelevant. Therefore, we may choose points A,B and C, as shown in the first accompanying figure, so that AB=2,BC=3 and (since AC=Ξ±+Ξ²)AC=4. Since β ACB=2Ξ±β, by the Law of Cosines we have
Attacking the problem more directly, let r denote the radius of the circle, and note that sin2Ξ±β=r1β, as shown in the second figure above. One similarly finds that sin2Ξ²β=2r3β and sin2Ξ±+Ξ²β=r2β. Since sin(2Ξ±β+2Ξ²β)=sin2Ξ±βcos2Ξ²β+cos2Ξ±βsin2Ξ²β, it follows that
r2β=r1β1β4r29ββ+2r3β1βr21ββ(1)
Solving this for r21β, one finds that it is equal to 6415β. Upon checking that this root is not extraneous to (1), it follows that
