Because triangle ABC is an equilateral triangle
So angle BAC= angle C=60 degrees.
AB=AC=BC
Because point D.E is the midpoint of AB and AC respectively.
So DE is the center line of triangle ABC.
So DE= 1/2BC
AD= 1/2AB
AE= 1/2AC
Parallel BC
So angle definition = angle EFC
So AD=AE=DE
So the triangle ADE is an equilateral triangle.
So angle ADE= angle ADQ+ angle EDG=60 degrees.
Because triangle DFG is an equilateral triangle
So DF=DG
Angle FDG= Angle EDF+ Angle EDG=60 degrees.
So angle ADG= angle EDF.
Because AD=DE (authentication)
DG=DF (certification)
So triangle DAG and triangle DEF are congruent (SAS)
So angle DAG= angle DEF
Because angle DAG= angle BAC+ angle CAG
Angle CAG=30 degrees
So the angle DAG=90 degrees
So EFC angle =90 degrees.
So CEF angle =90-60=30 degrees.
So CE= 1/2CF.
CE^2=EF^2+CF^2
Because CF= 1
So CE=2
So EF= root number 3