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8 points, 5 equal ratios given, angle between lines A_3B_3 , A_4B_4 wanted

Source: Sharygin 2005 finals 10.4

August 29, 2019
ratioequal ratioanglegeometry

Problem Statement

Two segments A1B1A_1B_1 and A2B2A_2B_2 are given on the plane, with A2B2A1B1=k<1\frac{A_2B_2}{A_1B_1} = k < 1. On segment A1A2A_1A_2, point A3A_3 is taken, and on the extension of this segment beyond point A2A_2, point A4A_4 is taken, so A3A2A3A1=A4A2A4A1=k\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k. Similarly, point B3B_3 is taken on segment B1B2B_1B_2 , and on the extension of this the segment beyond point B2B_2 is point B4B_4, so B3B2B3B1=B4B2B4B1=k\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k. Find the angle between lines A3B3A_3B_3 and A4B4A_4B_4.
(Netherlands)
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