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anemone
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Let $z_1=18+83i,\,z_2=18+39i$ and $z_3=78+99i$, where $i=\sqrt{-1}$. Let $z$ be the unique complex number with the properties that
$\dfrac{z_3-z_1}{z_2-z_1}\cdot \dfrac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.
$\dfrac{z_3-z_1}{z_2-z_1}\cdot \dfrac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.