= Solution
With $T$ from part (a), the first, second and fourth points map to $1,0,\infty$. The inverse image of the extended real line under a <Möbius transformation> is a <generalized circle>, meaning a circle or a line together with infinity. Thus, if $r=T(z_3)$ is real, all four original points lie on that generalized circle.
Conversely, if the four points lie on one <generalized circle>, its image under $T$ is another generalized circle containing $0,1,\infty$. A generalized circle containing infinity is a line, and the line through zero and one is the real line. Therefore $T(z_3)=r$ is real. Hence \b[the cross-ratio is real exactly when the four points are concyclic or collinear].
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