2017 AIME I Problem 10

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 \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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.