If three tangents to a circle form an equilateral triangle, prove that the tangent points form an equilateral triangle inscribed in the circle

I added a figure so you can guide yourself throughout the proof I'm about to write, so I recommend that you download the picture beforehand and have this window and the picture's window open. Alright, let's get started!
Assuming that is an equilateral triangle according to the wording of the problem, we have that the angles .
We also know that the circle in green is Inscribed in .

The following applies to every inscribed circle inside a triangle:
The center of the inscribed circle of a triangle is the intercept of all three angle bisectors of the triangle.

The above theorem implies that the line is an angle bisector because it goes through the vertex F and the center of the inscribed circle.

The previous statement implies that the angle .

Now let's work on the  triangle.
Knowing that (because  is tangent to the circle at B and  is a radius of the circle. If you're lost here, remember that a tangent to a circle is always perpendicular to the radius of the circle.) we can then derive that  (because the sum of the measures of all angles in a triangle is always equal to 180 degrees).

In the same way, we can prove also that:
.
Knowing the above we notice that .

We're at the last part of our proof here:
Now notice that  subtends the same arc on the circle  that .
According to the inscribed angle theorem, an angle  inscribed in a circle is half of the central angle  that subtends the same arc on the circle. 
Therefore 

We can prove in a similar fashion that:   
Therefore all the angles of the  triangle have a measure of 60 degrees, we conclude then that  is equilateral.