Question 10
Let $z_1 = 18 + 83\mathrm{i},$ $z_2 = 18 + 39\mathrm{i},$ and $z_3 = 78 + 99\mathrm{i},$ where $\mathrm{i} = \sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3 - z_1}{z_2 - z_1} \cdot \frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z.$