Suppose $z_1,z_2,z_3,z_4 \in \mathrm C$ lie on a generalized circle. $(z_1, z_2, z_3, z_4)$ is their cross ratio. Then $arg(z_1, z_2, z_3, z_4) = 0$ or $\pi$. My textbook uses this to show that this cross ratio is real (in fact, if the argument is either $0$ or $\pi$ then it is real). A seperate proof of the fact that the cross ratio of these points is real I found at https://math.stackexchange.com/a/932827/812647. I was thinking that the book is using Euler's formula somehow, but now I don't see how it could work. Any hints? Thanks in advance.
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1What is a generalized circle? – Gerry Myerson Mar 13 '22 at 09:09
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Is a "generalized circle" the same as either a circle or a line? – coffeemath Mar 13 '22 at 09:11
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@coffeemath yes – H-a-y-K Mar 13 '22 at 09:23
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Sorry, I had to omit the word generalized as the answer will suffice me in either case – H-a-y-K Mar 13 '22 at 09:25
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@JeanMarie I have already said that in my question. The book implies that property from the statement in my question, not the contrary. – H-a-y-K Mar 13 '22 at 10:07
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In fact the properties are equivalent... – Jean Marie Mar 13 '22 at 10:08
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1@JeanMarie yes, but the book is taking a different approach. It states the property in my question, to imply the property you are stating. Your approach first shows the second one to imply the first one. – H-a-y-K Mar 13 '22 at 10:10