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the time limit is up, here's a solution with no trig nor non-elementary formulae, but with heavy use of similar triangles. here d is the side-length of the rhombus and e is the length of the vertical side of the upper triangle. In the last picture, it is hard to see but the dark purple represents the outer triangle.
cc @Inginsub @Kazak @hazlin @theorytoe @ceo_of_monoeye_dating
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