To Prove the relation between HCF and LCM of three numbers

I decided to write my comment as an answer. Rather than start with naming $HCF(p,q)$, $HCF(q,r)$ and $HCF(r,p)$, start with $HCF(p,q,r)$. So let's call $HCF(p,q,r) = h$.

Next, write $HCF(p,q) = xh$, $HCF(q,r) = yh$ and $HCF(r,p) = zh$. It should be clear why we can assume the factor $h$ appears in all three, but you also know that $x,y,z$ are relatively prime. (Why?)

Thus, you can write $p = p'xzh$ for some $p'$, and similarly $q = q'xyh$ and $r = r'yzh$ (again, why?). What do you get when you plug those into your equation?