Find all tuples of positive integers $(a, b, c)$ such that lcm$(a, b, c)$ = $ab + bc + ca\over 4$
I am stuck on this problem for almost 2 hours. I tried the following things (however didn't reach the answer): 1)writing $a=dx,b=dy,c=dz$ and tried to proceed, but ended up making lots of cases number of cases, proceeding from that point seemed near impossible. 2)tried to directly make cases 3)Tried to bound by using i)AM-GM inequality ii)WLOG $a\ge b\ge c$, then $$c\le \text{lcm}(a,b,c)\le abc$$ but ultimately ended up with nothing.
Any help would be highly appreciated.