We have the following result:
If $a$, $b$ and $c$ are pairwise coprime integers, such that
$$\frac 1a+\frac 1b=\frac 1c$$
then $a+b$, $a-c$ and $b-c$ are perfect squares.
What I did.
I tried to prove first that $a+b$ is a perfect square.
Multiplying by $abc$ we get
Since $a$ and $c$ are coprime, we get by Gauss' lemma that
Since $a$ and $b$ are coprime,
But we obviously have $a+b\mid ab$, so
I think this would imply $a=b=2$ which is absurd since $a$ and $b$ are coprime...
The question. Is this going somewhere? What else could I do?