See the diagram.
This shows 1/3 of the proof.
Consider the part of the external triangle opposite angle A. Call angle A theta.
The diagram shows that the length for the part of the triangle opposite angle A is
2*R*tan(theta) = 2*R*tan(A)
Note: The 2*theta is measured from the center of the circle.
Repeating the procedure for the parts opposite B and C and summing, we get:
Perimeter = 2*R*tan(A) + 2*R*tan(B) + 2*R*tan(C) = 2R(tan(A) + tan(B) + tan(C))
Is this the formula you want to prove?