Question 132
$ABCD$ is a tetrahedron with the following property: letting $A'$, $B'$, $C'$, $D'$ be, respectively, the incenters of the faces $BCD$, $ACD$, $ABD$ and $ABC$, the lines $AA'$, $BB'$, $CC'$ and $DD'$ have a common point. Prove that the product of the lengths of two opposite edges of the tetrahedron is constant, that is, $AB \cdot CD = AC \cdot BD = AD \cdot BC$.