- #1
xman
- 93
- 0
Can some one help me out here, I'm stuck on the following problem:
show that four distinct points [tex] z_i , i=1,\ldots,4[/tex] in [tex] \hat{\mathbb C} [/tex] lie on a circle in [tex] \hat{\mathbb C} [/tex] if and only if the cross-ratio [tex] \left[ z_1,z_2,z_3,z_4 \right] [/tex] is real.
So, I know we can write the cross-ratio as
[tex] \left[z_1,z_2,z_3,z_4 \right] = \frac{(z_1 - z_3) (z_2-z_4)}{(z_1-z_2)(z_3-z_4)} [/tex]
I also know that I can make the problem simpler if I can reduce the problem to the case where
[tex] z_1=1,z_2=0,z_3=\infty [/tex]
but I have no idea even why if I could reduce the problem with to the above case, how that will help me. Can someone shed some light on this for me. Thanks in advance.
show that four distinct points [tex] z_i , i=1,\ldots,4[/tex] in [tex] \hat{\mathbb C} [/tex] lie on a circle in [tex] \hat{\mathbb C} [/tex] if and only if the cross-ratio [tex] \left[ z_1,z_2,z_3,z_4 \right] [/tex] is real.
So, I know we can write the cross-ratio as
[tex] \left[z_1,z_2,z_3,z_4 \right] = \frac{(z_1 - z_3) (z_2-z_4)}{(z_1-z_2)(z_3-z_4)} [/tex]
I also know that I can make the problem simpler if I can reduce the problem to the case where
[tex] z_1=1,z_2=0,z_3=\infty [/tex]
but I have no idea even why if I could reduce the problem with to the above case, how that will help me. Can someone shed some light on this for me. Thanks in advance.