Because the circle with the diameter of AB passes through BC at D and AC at E, connecting OD.

So angle CDE= angle BAC

Angle ADB=90 degrees

Angle DEC= angle b

Because AB=AC

So triangle ABC is an isosceles triangle

Angle ABC= angle ACB

AD is the center line and angular bisector of isosceles triangle ABC.

So angle BAD= 1/2 angle BAC.

BD=DC

So angle DEC= angle ACB

So de =DC

So the triangle DEC is an isosceles triangle.

Because DF is perpendicular to AC

So the angle CDF= 1/2 angle CDE.

So angle BAD= angle CDF

Because angle ABC+ angle BAD+ angle ADB= 180 degrees.

So angle ABC+ angle CDF=90 degrees.

Because OB=OD

So angle ABC= angle ODB

So angle ODB+ angle CDF=90 degrees.

Because angle ODB+ angle ODF+ angle CDF= 180 degrees.

So ODF angle =90 degrees

Because OD is the radius of circle o.

So DF is the tangent of circle o.

BD=DE

] so arc BD= arc DE

So AE=2EF is not necessarily correct.