How can I show that $\forall$ distinct $z_1,z_2,z_3,z_4 \in \mathbb{C}$, $\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} \in \mathbb{R} \iff$ the points lie on a circle? I don't really have a clue of the first step
2026-05-16 14:30:17.1778941817
Circleline points
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Hint : Multiply by $$\frac{(\bar{z_1}-\bar{z_4})(\bar{z_2}-\bar{z_3})}{(\bar{z_1}-\bar{z_4})(\bar{z_2}-\bar{z_3})}$$ to get a real denominator. Then equal the imaginary part on the numerator to $0$.