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equilateral wanted, triangle 30-75-75 given

Source: Norwegian Mathematical Olympiad 1999 - Abel Competition p3

February 11, 2020
isoscelesEquilateraltriangle areaarea

Problem Statement

An isosceles triangle ABCABC with AB=ACAB = AC and ∠A=30o\angle A = 30^o is inscribed in a circle with center OO. Point DD lies on the shorter arc ACAC so that ∠DOC=30o\angle DOC = 30^o, and point GG lies on the shorter arc ABAB so that DG=ACDG = AC and AG<BGAG < BG. The line BGBG intersects ACAC and ABAB at EE and FF, respectively. (a) Prove that triangle AFGAFG is equilateral. (b) Find the ratio between the areas of triangles AFEAFE and ABCABC.
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