Question 19
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
$$ \frac{DB}{DD^{\prime}}+\frac{EC}{EE^{\prime}}+\frac{FA}{FF^{\prime}}. $$
$$ \angle ADC=2\angle BAC,\quad\angle BEA=2\angle ABC,\quad\angle CFB=2\angle ACB. $$
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
$$ \frac{DB}{DD^{\prime}}+\frac{EC}{EE^{\prime}}+\frac{FA}{FF^{\prime}}. $$