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path by equal angles across a semicircle , billiard type

Source: Spanish Mathematical Olympiad 1967 P5

December 4, 2022
geometryequal angles

Problem Statement

Let γ\gamma be a semicircle with diameter ABAB . A creek is built with origin in AA , which has its vertices alternately in the diameter ABAB and in the semicircle γ\gamma , so that its sides make equal angles α\alpha with the diameter (but alternately in either direction). It is requested: a) Values of the angle α\alpha for the ravine to pass through the other end BB of the diameter. b) The total length of the ravine, in the case that it ends in BB , as a function of the length dd of the diameter and of the angle α\alpha . https://cdn.artofproblemsolving.com/attachments/3/c/54c71cdf1bf8fbb3bdfa38f4b14a1dc961c5fe.png
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