On second thought, your way works.
|z1|=|z2|=|z3|=1 and z1+z2+z3 = 1 does imply a rhombus with vectors z1,z2,z3, and (1,0).
Since a rhombus has parallel sides, one of z1/z2/z3 has to be the vector (1,0).
Not completely sure if you'll get the marks since it's HSC marking, but this method does...