ILL 1971 Problem 32

Two half-lines $a$ and $b$ , with the common endpoint $O$ , make an acute angle $\alpha$ . 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 $\beta$ be the circle with centre $B$ and radius $BO$ . We construct a sequence of half-lines $c_1,c_2,c_3,\ldots$ , all lying inside the angle $\alpha$ , in the following manner: (i) $c_i$ is given arbitrarily; (ii) for every natural number $k$ , the circle $\beta$ intercepts on $c_k$ a segment that is of the same length as the segment cut on $b'$ by $a$ and $c_{k+1}$ . Prove that the angle determined by the lines $c_k$ and $b$ has a limit as $k$ tends to infinity and find that limit.