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sum of ratios wanted, similar isosceles triangles on sides of a triangle

Source: 2019 Dürer Math Competition Finals E1.4

November 28, 2020
ratiogeometryisosceles

Problem Statement

Let ABCABC be an acute-angled triangle having angles α,β,γ\alpha,\beta,\gamma at vertices A,B,CA, B, C respectively. Let isosceles triangles BCA1,CAB1,ABC1BCA_1, CAB_1, ABC_1 be erected outwards on its sides, with apex angles 2α,2β,2γ2\alpha ,2\beta ,2\gamma respectively. Let A2A_2 be the intersection point of lines AA1AA_1 and B1C1B_1C_1 and let us define points B2B_2 and C2C_2 analogously. Find the exact value of the expression AA1A2A1+BB1B2B1+CC1C2C1\frac{AA_1}{A_2A_1}+\frac{BB_1}{B_2B_1}+\frac{CC_1}{C_2C_1}
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