Let \( z_{1}, z_{2} \) and \( z_{3} \) represent the vertices \( A, B \) and \( C \) of the tria...

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Let \( z_{1}, z_{2} \) and \( z_{3} \) represent the vertices \( A, B \) and \( C \) of the triangle \( A B C \), respectively, in the Argand plane, such that \( \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=5 \). Prove that \( z_{1} \sin 2 A+z_{3} \sin 2 B+z_{3} \sin \) \( 2 C=0 \).
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