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Math Prize 2014 Problem 17

Source:

September 29, 2014
geometryratioquadraticsanalytic geometryarea of a trianglealgebraquadratic formula

Problem Statement

Let ABCABC be a triangle. Points DD, EE, and FF are respectively on the sides BC‾\overline{BC}, CA‾\overline{CA}, and AB‾\overline{AB} of △ABC\triangle ABC. Suppose that AEAC=CDCB=BFBA=x \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x for some xx with 12<x<1\frac{1}{2} < x < 1. Segments AD‾\overline{AD}, BE‾\overline{BE}, and CF‾\overline{CF} cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of xx.
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