Q:

Is the following substitution legal? 

I have to prove that from 
$a^3+b^3+c^3=3abc$ and $a+b+c \neq0$
it follows that 
$4b^2-4ac=0$.
Am I allowed to substitute $ac=b^2$ in $a^3+b^3+c^3=3abc$ and if i prove the resulting equality is true then it will follow that $ac=b^2$ because otherwise the resulting equality wouldn't be true?? 

A:

Since your actual question concerns the legality of a substitution, my answer is a simple no.  The reason is because this is a good example of a backwards proof.  You are actually assuming what you are trying to prove.  You want to show a statement of the form $p\to q$, but what you are doing is trying to obtain something of the form $(q\land p)\to r$, where $r$ is some true equality.  This does not really constitute a valid proof.  You need to work toward the statement you are trying to prove true, not from it. 