Paper 3, Section I, E

Groups | Part IA, 2017

Let w1,w2,w3w_{1}, w_{2}, w_{3} be distinct elements of C{}\mathbb{C} \cup\{\infty\}. Write down the Möbius map ff that sends w1,w2,w3w_{1}, w_{2}, w_{3} to ,0,1\infty, 0,1, respectively. [Hint: You need to consider four cases.]

Now let w4w_{4} be another element of C{}\mathbb{C} \cup\{\infty\} distinct from w1,w2,w3w_{1}, w_{2}, w_{3}. Define the cross-ratio [w1,w2,w3,w4]\left[w_{1}, w_{2}, w_{3}, w_{4}\right] in terms of ff.

Prove that there is a circle or line through w1,w2,w3w_{1}, w_{2}, w_{3} and w4w_{4} if and only if the cross-ratio [w1,w2,w3,w4]\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 C{}.]\mathbb{C} \cup\{\infty\} .]

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