Triangle inequalities

[ins-]

It is given acute angled triangle $ABC$. Let $H_1, H_2, H_3$ are the feets of the altitudes, $T_1, T_2, T_3$ are the tangent points of incircle with the sides, $L_1, L_2, L_3$ are the intersection points of the internal angle bisectors with the sides. Prove that: $S_{H_1H_2H_3} \leq S_{T_1T_2T_3} \leq S_{L_1L_2L_3} \leq \frac{1}{4}S_{ABC}$.

[ins-]

(Оффтоп)

Here I'm providing some facts that may make the problem easier:
$\frac{S_{H_1H_2H_3}}{S_{ABC}}=2cosAcosBcosC$
$\frac{S_{T_1T_2T_3}}{S_{ABC}}=\frac{r}{2R}$
$\frac{S_{L_1L_2L_3}}{S_{ABC}}=\frac{2mnp}{(m+n)(n+p)(p+m)}$
where $A,B,C$ are the triangle's angles, $r$ is inradii, $R$ is circumradii, and the trianges sides are in ratio $m:n:p$ $(a:b:c=m:n:p)$.

[ins-]

(Оффтоп)

If you are interested you can see the solution on the following link
http://www.artofproblemsolving.com/Foru ... 4#p2664884