Problem 1746 (difficulty: 4/10)

(a) Prove that \(\displaystyle \displaystyle(z_1, z_2, z_3):= \frac{z_1 - z_3}{z_2 - z_3}\) is real if and only if \(\displaystyle z_1\), \(\displaystyle z_2\) and \(\displaystyle z_3\) are on a line.

(b) Prove that the cross-ratio \(\displaystyle \displaystyle(z_1, z_2, z_3, z_4):= \frac{z_1 - z_3}{z_2 - z_3} : \frac{z_1 - z_4}{z_2 - z_4}\) is real if and only if \(\displaystyle z_1\), \(\displaystyle z_2\), \(\displaystyle z_3\) and \(\displaystyle z_4\) are on a circline.

Give me another random problem!