You are asked to take three distinct points $1 , \omega _ { 1 }$ and $\omega _ { 2 }$ in the complex plane such that $\left| \omega _ { 1 } \right| = \left| \omega _ { 2 } \right| = 1$. Consider the triangle T formed by the complex numbers $1 , \omega _ { 1 }$ and $\omega _ { 2 }$. Statements (5) There is exactly one such triangle T that is equilateral. (6) There are exactly two such triangles $T$ that are right angled isosceles. (7) If $\omega _ { 1 } + \omega _ { 2 }$ is real, the triangle T must be isosceles. (8) For any nonzero complex number $z$, the numbers $z , z \omega _ { 1 }$ and $z \omega _ { 2 }$ form a triangle that is similar to the triangle T.
You are asked to take three distinct points $1 , \omega _ { 1 }$ and $\omega _ { 2 }$ in the complex plane such that $\left| \omega _ { 1 } \right| = \left| \omega _ { 2 } \right| = 1$. Consider the triangle T formed by the complex numbers $1 , \omega _ { 1 }$ and $\omega _ { 2 }$.
\textbf{Statements}
(5) There is exactly one such triangle T that is equilateral.\\
(6) There are exactly two such triangles $T$ that are right angled isosceles.\\
(7) If $\omega _ { 1 } + \omega _ { 2 }$ is real, the triangle T must be isosceles.\\
(8) For any nonzero complex number $z$, the numbers $z , z \omega _ { 1 }$ and $z \omega _ { 2 }$ form a triangle that is similar to the triangle T.