$a,b,c$ are complex numbers whose modulus are equal. If $a+bc,b+ca,c+ab$ are real then find the value of $abc$?
We can assume three complex numbers and then make six equations but that seems a very tedious process. Is there any smart approach to do this question?
I was also thinking if we take all three numbers as 0 then all conditions are satisfied but I am not sure if that is the way to go.
NOTES:
One family of solutions to your equations is to take any real number $k$ and take
$$a=\pm k,\quad b=\pm k,\quad c=\pm k$$
where each plus-or-minus is independent of the others. Then we get
$$abc=\pm k^3$$
This means $abc$ can be any real number. Thus there is no unique value of $abc$. There may be other families of solutions where $abc$ is not real, of course, so this is not a complete solution. But it does show that at best we can get curves of the possible values of $abc$. Could this be what your problem is asking?
We can reduce the number of variables involved in the problem by letting
$$a=re^{si}, \quad b=re^{ti}, \quad c=re^{ui}$$
Here $r$ is the common modulus, and $r,s,t,u$ are real numbers. Your equations then give three equations in four variables. We can expect just one degree of freedom in the solutions, so we expect to get one or more curves in the plane as our solution set. We already found one curve: there may be others.
