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I have to prove that sum of the angles in triangle is equal to $\pi$ using complex numbers. Can anyone give me some hints on how to do it? Thank you!

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Hint: If the vertices of the triangle are represented by complex numbers $\ z_1\ $, $\ z_2\ $ and $\ z_3\ $, and the angles at these vertices are $\ \theta_1\ $, $\ \theta_2\ $, and $\ \theta_3\ $, respectively, then \begin{align} \frac{z_2-z_1}{\left\vert z_2-z_1\right\vert}&=e^{i\theta_1}\frac{z_3-z_1}{\left\vert z_3-z_1\right\vert}\ ,\\ \frac{z_3-z_2}{\left\vert z_3-z_2\right\vert}&=e^{i\theta_2}\frac{z_1-z_2}{\left\vert z_1-z_2\right\vert}\ , \text{and}\\ \frac{z_1-z_3}{\left\vert z_1-z_3\right\vert}&=e^{i\theta_3}\frac{z_2-z_3}{\left\vert z_2-z_3\right\vert} .\\ \end{align} What happens if you multiply these equations together?

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