KöMaL 2018 B4956

A transformation of central similitude is applied to a tetrahedron $ABCD$ with each vertex as centre. The four diminished tetrahedra obtained are $AA_bA_cA_d$, $B_aBB_cB_d$, $C_aC_bCC_d$ and $D_aD_bD_cD$. Given that these small tetrahedra are pairwise disjoint, prove that the volumes of tetrahedra ${A_b B_c C_d D_a}$, ${A_b B_d D_c C_a}$, ${A_c C_b B_d D_a}$, ${A_c C_d D_b B_a}$, ${A_d D_b B_c C_a}$ and ${A_d D_c C_b B_a}$ are equal.