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.