Two half-lines a and b, with the common endpoint O, make an acute angle α. Let A on a and B on b be points such that OA=OB, and let b be the line through A parallel to b. Let β be the circle with centre B and radius BO. We construct a sequence of half-lines c1,c2,c3,…, all lying inside the angle α, in the following manner:
(i) ci is given arbitrarily;
(ii) for every natural number k, the circle β intercepts on ck a segment that is of the same length as the segment cut on b′ by a and ck+1.
Prove that the angle determined by the lines ck and b has a limit as k tends to infinity and find that limit. geometry proposedgeometry