Let z1 and z2 be two non-zero complex numbers such that z1z2+z2z1=1, then the origin and points represented by z1 and z2

Let z=z1z2, and z+1z=1⇒ z2-z+1=0⇒ z=1±3i2 ⇒ z1z2=1±3i2If z1 and z2 are represented by A and B respectively and O be the origin, thenOAOB=z1z2=1±3i2=14+34=1⇒ OA=OBAlso, ABOB=z2−z1z2=1−z1z2     =1−12±32i=12∓32i=14+34=1⇒  AB=OBThus, OA=OB=AB.∴ ∆OAB is an equilateral triangle.