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Given three points $\frac{3}{2} + i , 2i,-6+6i$. I have the mobius transformation that maps these three points to $0,1,\infty$ respectively as

$M(z) = \frac{(-4i+6)(z-(1+2i))}{(3-7i)(z-(10-20i)}$

My textbook included the method to determine whether four points lie on a generalized circle but didn't show how to determine if they lie on the same line. I am not being able to get my head around it and come up with the method to determine whether these three points lie on the same line using Mobius transformation. Thank you very much for your response and help.

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  • $\begingroup$ Does this transformation maps $3/2+i$ to 0 and maps $-6+6i$ to $\infty$ as desired? For your second question, you may try the method of undetermined coefficients. $\endgroup$ Commented May 21, 2019 at 1:04
  • $\begingroup$ Use the same method but use infinity for the fourth point. A generalized cirlce with one point at infinity is the same as a line. $\endgroup$ Commented May 21, 2019 at 2:51

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