NL-NWO 1991 Problem 2

An angle with vertex $A$ and measure $\alpha$ and a point $P_0$ on one of its rays are given so that $AP_0=2$ . Point $P_1$ is chose on the other ray. The sequence of points $P_1,P_2,P_3,...$ is defined so that $P_n$ lies on the segment $AP_{n-2}$ and the triangle $P_n P_{n-1} P_{n-2}$ is isosceles with $P_n P_{n-1}=P_n P_{n-2}$ for all $n \ge 2$ . $(a)$ Prove that for each value of $\alpha$ there is a unique point $P_1$ for which the sequence $P_1,P_2,...,P_n,...$ does not terminate. $(b)$ Suppose that the sequence $P_1,P_2,...$ does not terminate and that the length of the polygonal line $P_0 P_1 P_2 ... P_k$ tends to $5$ when $k \rightarrow \infty$ . Compute the length of $P_0 P_1$ .

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