Let $ABC $ be an acute-angled triangle. Let $D$, $E$, $F$ be the feet of the perpendiculars from $A $, $B$, $C $ on the opposite sides $BC $, $CA $, $AB $. Let $\rho,\rho_1,\rho_2 ,\rho_3$ be the radii of the circles inscribed in the triangles $DEF$, $AEF$, $BFD$, $CDE$.
Prove that $r^3\rho=2R \rho_1\rho_2 \rho_3$.
Here, $r $ is the radius of the incircle of triangle $ABC$, and $R$ is the circumradius of $ ABC$.