Question 3
Prove that for every point $M$ inside triangle $ABC$,
$$d_ad_bd_c\le\frac{8S^3}{27abc},$$
where $d_a,d_b,d_c$ are the distances from $M$ to sides $BC,CA,AB$, respectively; $a,b,c$ are their lengths and $S$ is the area of the triangle. Extend this inequality to a tetrahedron in space.
$$d_ad_bd_c\le\frac{8S^3}{27abc},$$
where $d_a,d_b,d_c$ are the distances from $M$ to sides $BC,CA,AB$, respectively; $a,b,c$ are their lengths and $S$ is the area of the triangle. Extend this inequality to a tetrahedron in space.