MathDB
compute the length

Source: Netherlands 1991

June 28, 2009
geometry proposedgeometry

Problem Statement

An angle with vertex A A and measure α \alpha and a point P0 P_0 on one of its rays are given so that AP_0\equal{}2. Point P1 P_1 is chose on the other ray. The sequence of points P1,P2,P3,... P_1,P_2,P_3,... is defined so that Pn P_n lies on the segment AP_{n\minus{}2} and the triangle P_n P_{n\minus{}1} P_{n\minus{}2} is isosceles with P_n P_{n\minus{}1}\equal{}P_n P_{n\minus{}2} for all n≥2 n \ge 2. (a) (a) Prove that for each value of α \alpha there is a unique point P1 P_1 for which the sequence P1,P2,...,Pn,... P_1,P_2,...,P_n,... does not terminate. (b) (b) Suppose that the sequence P1,P2,... P_1,P_2,... does not terminate and that the length of the polygonal line P0P1P2...Pk P_0 P_1 P_2 ... P_k tends to 5 5 when k→∞ k \rightarrow \infty. Compute the length of P0P1 P_0 P_1.
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