Four circles are tangent to each other and to a triangle that surrounds them, as shown in the figure. Each of the three circles in the corners have a radius of 1 cm. What is the radius of the circle in the middle?

First, I like your proof. It's a really nice proof. Second, I checked everything except the final calculation of R and it seems to have no errors. Except that triangles ABF and DBF are congruent, not similar.