The incircle $T$ of the scalene $\triangle ABC$ touches $BC$ at $D$, $CA$ at $E$ and $AB$ at $F$. lf $R_1$ be the radius of the circle inside $\triangle ABC$ which is tangent to $T$ and the sides $AB$ and $AC$. Define $R_2$ and $R_3$ similarly. If $R_1=16$, $R_2=25$, $R_3=36$, determine radius of $T$.
My Effort I tried drawing diagram but felt completely clue-less.
It's quite easy to show that if $R$ is the radius of $T$, $$\sin\frac A2=\frac{R-16}{R+16}$$ $$\sin\frac B2=\frac{R-25}{R+25}$$ $$\sin\frac C2=\frac{R-36}{R+36}$$ We may now use the following identity since $A+B+C=\pi$: $$\sin^2\frac A2+\sin^2\frac B2+\sin^2\frac C2+2\sin\frac A2\sin\frac B2\sin\frac C2=1$$ Substituting the expressions for $\sin\frac X2$ in terms of $R$ and solving the resulting equation yields the only positive solution as $R=74$.
More generally, if $R_1=a^2$, $R_2=b^2$ and $R_3=c^2$ then $R=ab+bc+ca$.