I have a triangle $ABC$ with inradius $r$. Points $D$, $E$, $F$ are chosen on side $BC$, $CA$, $AB$, respectively, such that $\triangle AFE$, $\triangle BDF$, and $\triangle CED$ have same inradius $r_1$. Compute the inradius of $\triangle DEF$ in terms of $r$ and $r_1$.
I have an approach in mind by taking the various side lengths as x,y,z and so on but I know for sure it will turn out to be lengthy. Any better method please?