If the cross ratio $Z_1, Z_2, Z_3$ and $Z_4$ is real, then
which of the following statement is true?
1)$Z_1, Z_2$ and $Z_3$ are collinear
2)$Z_1, Z_2$ and $Z_3$ are concyclic
3)$Z_1, Z_2$ and $Z_3$ are collinear when atleast one $Z_1, Z_2$ or $Z_3$ is real
My attempt : By theorem: Cross ratio is real on image of real axis
I am confused due to $Z_4$ is not being given. Please help me.
As the answer in your link describes, if the cross-ratio of $z_1$, $z_2$, $z_3$ and $z_4$ is real, then the four points either lie on a straight line or a circle.
If all four points line on a straight line or a circle, then the same will be true for any three points chosen from them. So in answer to the question, either 1) or 2) will be true. There is no requirement for any individual point to be real so 3) is not necessarily true. It is certainly the case that one of points could be real but the set as a whole could line on a circle.
Whenever you have three distinct points on the plane, they are either collinear or cocyclic, with no extra hypothesis. So, 1) or 2) is true, but you cannot decide between them.
Statement 3) can also be true or false. It depends.
It is indeed a strange exercise.