The worksheet below includes a sample use of the Mobius transformation which produces a hyperbolic reflection of a point in the Poincare disk, followed by my attempt to produce the same result from first principles in a procedure.

Do I misunderstand the Mobius transformation and its use and/or is my procedure incorrect? 

InversePoint.mw

Is there a way to directly code a complex conjugate such as z with overbar without using the verbose conjugate(z)?

This worksheet contains an unnamed theorem on page 202 of David Wells's book The Penguin Dictionary of Curious and Interesting Geometry.

Somehow I have uploaded both its contents and (a) link(s) to it.

What Maple code can animate and display, in turn, each of the portrayed pursuit paths?

Pursuit_problem.mw

Download Pursuit_problem

1) the two cylinders are centered on the x and z axis respectively

2) any two intersecting cylinders

As an example, the second display in the web site below shows the 42 possible triangulations of a cyclic heptagon polygon.

https://en.wikipedia.org/wiki/Polygon_triangulation