Let A, B, C, D represent 1, z1, z2, z1z2 respectively.
In triangles OAB, OCD:
angle OAB = angle OCD (arg(z1z2) - arg(z2) = arg(z1) - arg(1))
OB/OA = |z1|/|1| = OD/OC = |z1z2|/|z2|
hence OAB similar to OCD (equal angle between two pairs of sides of equal ratio.