2001 IMO Shortlist Problems/G5

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Problem

Let $ABC$ be an acute triangle. Let $DAC,EAB$ , and $FBC$ be isosceles triangles exterior to $ABC$ , with $DA = DC, EA = EB$ , and $FB = FC$ , such that

$\angle ADC = 2\angle BAC, \quad \angle BEA = 2 \angle ABC, \quad \angle CFB = 2 \angle ACB.$

Let $D'$ be the intersection of lines $DB$ and $EF$ , let $E'$ be the intersection of $EC$ and $DF$ , and let $F'$ be the intersection of $FA$ and $DE$ . Find, with proof, the value of the sum