Erdős-Mordell theorem geometry proof 
Using the notation of the Erdős-Mordell theorem, prove that $PA \cdot PB \cdot PC \geq \dfrac{R}{2r}(p_a+p_b)(p_b+p_c)(p_c+p_a)$.

The notation of the Erdős Mordell theorem means that $p_a$ for example is the distance from the point $p$ to the side $a$ and $R$ is the circumradius. I am struggling to see how to use the product $PA \cdot PB \cdot PC$. We also have to relate this somehow to the circumradius. 
EDIT: Sorry there was a typo originally. The $24$ should've been a $2r$.

To complement the nice solution of @DeepSea, here is a place where one can find a proof of formula $r = 4R\sin\left(\dfrac{A}{2}\right)\sin\left(\dfrac{B}{2}\right)\sin\left(\dfrac{C}{2}\right)$ (and many others as well): 
http://www.cut-the-knot.org/triangle/RelationsInTriangle.shtml
