Mathematical equilateral triangle

Solution: connect DE

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