The incircle $$\Gamma$$ of a scalene triangle $$ABC$$ touches $$BC$$ at $$D$$, $$CA$$ at $$E$$ and $$AB$$ at $$F$$. Let $$r_A$$ be the radius of the circle inside $$ABC$$ which is tangent to $$\Gamma$$ and the sides $$AB$$ and $$AC$$. Define $$r_B$$ and $$r_C$$ similarly. If $$r_A = 16$$, $$r_B = 25$$ and $$r_C = 36$$, determine the radius of $$\Gamma$$.

The small circle in the corner $$A$$ is the image of $$\Gamma$$ under a homothety at $$A$$, which gives $$r_A = r \cdot \frac{1 - \sin\frac{A}{2}}{1 + \sin\frac{A}{2}}$$, and similarly for $$r_B$$ and $$r_C$$. Combining these with $$\frac{A}{2} + \frac{B}{2} + \frac{C}{2} = 90^\circ$$ yields the identity $$r = \sqrt{r_A r_B} + \sqrt{r_B r_C} + \sqrt{r_C r_A}$$. Substituting the given values gives $$r = 20 + 30 + 24 = 74$$.