Paper 3, Section I, E

Let $w_{1}, w_{2}, w_{3}$ be distinct elements of $\mathbb{C} \cup\{\infty\}$. Write down the Möbius map $f$ that sends $w_{1}, w_{2}, w_{3}$ to $\infty, 0,1$, respectively. [Hint: You need to consider four cases.]

Now let $w_{4}$ be another element of $\mathbb{C} \cup\{\infty\}$ distinct from $w_{1}, w_{2}, w_{3}$. Define the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ in terms of $f$.

Prove that there is a circle or line through $w_{1}, w_{2}, w_{3}$ and $w_{4}$ if and only if the cross-ratio $\left[w_{1}, w_{2}, w_{3}, w_{4}\right]$ is real.

[You may assume without proof that Möbius maps map circles and lines to circles and lines and also that there is a unique circle or line through any three distinct points of $\mathbb{C} \cup\{\infty\} .]$

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