Prove that the cross-ratio of four complex numbers, defined as

is real if and only if the four points lie on a circle.

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- Jun 8th 2010, 04:55 PMBruno J.Cross-ratio
Prove that the cross-ratio of four complex numbers, defined as

is real if and only if the four points lie on a circle.

- Jun 8th 2010, 05:38 PMwonderboy1953To clarify this problem
I presume you mean the circumference and not inside of it.

- Jun 8th 2010, 05:40 PMBruno J.
A circle is the boundary (circumference) of a disc.

- Jun 8th 2010, 08:18 PMtonio
- Jun 8th 2010, 08:21 PMBruno J.
Good job!

Here's my solution:

Let be given by , where are any three distinct real points.

Then is a Möbius transformation, which takes circles to circles. The circle containing is therefore mapped to the real line, since the three points are mapped to the real line.

(This is essentially equivalent to your second solution!) - Jun 8th 2010, 09:47 PMsimplependulum
- Jun 8th 2010, 10:31 PMBruno J.