lower and upper bounds on $aA+ bB + cC$ and ${\large{\frac{ab}{l_c}}}+{\large{\frac{bc}{l_a}}}+ {\large{\frac{ca}{l_b}}}$.

Question

Let $ABC$ be an acute triangle.

The goal is to prove: \begin{align*} &(a)\;\;\;\pi(2R-r) < aA+ bB + cC < 4(2R-r)\\[2pt] &\qquad\qquad\text{[where in the above, $A,B,C$ denote the respective radian measures]}\\[6pt] &(b)\;\;\;\frac{12R}{\pi} < \frac{ab}{l_c}+ \frac{bc}{l_a}+ \frac{ca}{l_b} < 3\pi R\\[4pt] \end{align*} where

• $a,b,c$ are the lengths of the sides opposite vertices $A,B,C$, respectively.$\\[4pt]$
• $p$ is the perimeter.$\\[4pt]$
• $r,R$ are the inradius and circumradius, respectively.$\\[4pt]$
• $l_a, l_b,l_c$ are the lengths of the angle bisectors of angles $A,B,C$, respectively.