Let
be an acute triangle. Let
, and
be isosceles triangles exterior to
, with
, and
, such that
Let
be the intersection of lines
and
, let
be the intersection of
and
, and let
be the intersection of
and
. Find, with proof, the value of the sum
%V0
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
$$\frac{DB}{DD'}+\frac{EC}{EE'}+\frac{FA}{FF'}.$$
Školjka