Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Prove it Estonia

In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36°$ . Find all possibilities, how big the angles of this triangle can be.

Solution

Based on the initial conditions the angles of the triangle are $\alpha$ , $2\alpha$ and $180° - 3\alpha$ and they all have to be different. It remains to perform calculations for three cases: $\alpha = 36°$ , $2\alpha = 36°$ and $180° - 3\alpha = 36°$ .

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