Can two different triads of natural numbers give the same results to these operations? If I have a triad of natural numbers $a$, $b$ and $c$, and another triad $a'$, $b'$ and $c'$, such that either $a\neq a'$, $b\neq b'$, or $c\neq c'$, is it possible that these equations are true?
$ab+c=a'b'+c'$
$ac+b=a'c'+b'$
$cb+a=c'b'+a'$
If I had only the first equation, it is obviously possible, for example $a=8$, $b=2$ and $c=14$ give the same result as $a'=10$, $b'=2$ and $c'=10$, when we introduce the other two equations it feels like it would be impossible but I don't know how to prove it

There are (perhaps some might say trivial) counterexamples of the form $(1,m,n)$ and $(1,n,m)$--for instance $a=1,b=3,c=4$ and $a'=1, b'=4, c'=3$
There are also nontrivial counterexamples such as:
$$a=2,b=5,c=89 {\rm \;\; and\;\; } a'=5,b'=13, c'=34$$
I found this with a search using Python.
I just noticed: They are all Fibonacci numbers.
Here's one that isn't all Fibonacci numbers:
$$a=2,b=9,c=67 {\rm \;\; and\;\; } a'=3,b'=14, c'=43$$
