ELMO 2021 Problem 6

In $\triangle ABC$ , points $D$ , $E$ , and $F$ lie on sides $BC$ , $CA$ , and $AB$ , respectively, such that each of the quadrilaterals $AFDE$ , $BDEF$ , and $CEFD$ has an incircle. Prove that the inradius of $\triangle ABC$ is twice the inradius of $\triangle DEF$ .