In a triangle ABC, the excircle E is tangent to the side BC at T (see the figure below). D and F are the incenters of triangles ABT and ATC. If A1, A2, A3, A4 are the areas of triangles BDT, TFC, BTE, and CTE, prove that \(\dfrac{A_{1}}{A_{2}}=\dfrac{{A_{3}}^{2}}{{A_{4}}^{2}}\).

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Ten problems: 1411-1420

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