Complez number problem

Hi guys,
I'm really stuck on a complex number problem.

We have: z1 and z2 are two complex numbers. |z1|=25  |z1+z2|=25 |z1-z2|=25
Find what is z2 (all the possibilities)

We can easily see that z2=0 is a solution.

I tried to develop the norm of each term and trying to reduce them, but I can't find any answer please help me!
Gabriel

z1 = a+bi
z2 = c+di
|z1| = a^2+b^2 = 25
|z1+z2|= (a+c)^2+(b+d)^2 = a^2 +2ac +c^2 +b^2 +2bd +d^2 = 25
2ac +c^2 +2bd +d^2 = 0
|z1-z2|= (a-c)^2+(b-d)^2 = a^2 -2ac +c^2 +b^2 -2bd +d^2 = 25
-2ac +c^2 -2bd +d^2 = 0

2ac +c^2 +2bd +d^2 = -2ac +c^2 -2bd +d^2
2ac +2bd = -2ac -2bd
2ac +2bd = -(2ac +2bd)
2ac + 2bd = 0
c^2 +d^2 = 0
c and d are reals, so c^2 and d^2 cant be negative, the only solution is c=0,d=0
z2=0