2017 AIME I Problem 10

Let z_{1}=18+83 i, z_{2}=18+39 i, and z_{3}=78+99 i, 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.