Complex Numbers

Given the complex numbers z1,z2,z3 with unit modulus, we know:
z1z1=z2z2=z3z3=1
This implies that each complex number has a magnitude of 1 . We are also given the condition:
|z1−z2|2+|z1−z3|2=4
We can expand the squared magnitudes as follows:
‌|z1−z2|2=(z1−z2)(z1−z2)=z1z1−z1z2−z1z2+z2z2,
‌|z1−z3|2=(z1−z3)(z1−z3)=z1z1−z1z3−z1z3+z3z3.
Substitute the expansions back into the equation:
‌(z1z1−z1z2−z1z2+z2z2)
‌+(z1z1−z1z3−z1z3+z3z3)=4.
Since z1z1=z2z2=z3z3=1, we simplify:
(1−z1z2−z1z2+1)+(1−z1z3−z1z3+1)=4
Simplifying further gives:
4−(z1z2+z1z2+z1z3+z1z3)=4
Thus:
z1z2+z1z2+z1z3+z1z3=0
This is the required result.