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Combinatorics

Source: Archimedes Junior 2012

March 16, 2020
combinatorics

Problem Statement

On a plane Π\Pi is given a straight line ℓ\ell and on the line ℓ\ell are given two different points A1,A2A_1, A_2. We consider on the plane Π\Pi, outside the line ℓ\ell, two different points A3,A4A_3, A_4. Examine if it is possible to put points A3A_3 and A4A_4 on such positions such the four points A1,A2,A3,A4A_1, A_2, A_3, A_4 form the maximal number of possible isosceles triangles, in the following cases: (a) when the points A3,A4A_3, A_4 belong to dierent semi-planes with respect to ℓ\ell; (b) when the points A3,A4A_3, A_4 belong to the same semi-planes with respect to ℓ\ell. Give all possible cases and explain how is possible to construct in each case the points A3A_3 and A4A_4.
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