The complex numbers z1, z2 and z3 satisfying z1−z3z2−z3=12(1−3i) are vertices of a triangle which is

z1−z3z2−z3=z1−z3z2−z3=1−i32=14(1+3)=1⇒ z1−z3=z2−z3Also, z1−z3z2−z3−1=1−i32−1→ z1−z2z2−z3=−1−i32⇒ z1−z2z2−z3=14(1+3)=1⇒ z1−z2=z2−z3Thus , z1−z3=z2−z3=z2−z1Hence, z1, z2 and z3 are the vertices of an equilateral triangle.