This lab is just to help you make a Euclidean script that can be used to construct the hyperbolic mirror (perpendicular bisector) that reflects (hyperbolically speaking) one given point to another.

The following explains the reasoning behind the method of constructing a tangent from an external point:

For a given point external to a circle, it would be nice to construct the points of tangency from the point to the circle. (This would give a way to construct a hyperbolic line, if the corresponding Euclidean circle center were known.)	But how to find those points of tangency? One way is derived from thinking about the right triangle AXP, which we also of course can't construct.
But we could construct a copy of triangle AXP, AX'P', by extending AX to AP', so that AP'=AP, and then reflecting X in the bisector of angle PAP' to construct point X'. (Why must X' be on the circle and on AP?)	Of course, all that assumes that we already have point X. But we can sort of reverse the process, because we can construct X' as the intersection of AP with the circle ....
construct the perpendicular to AP through X' ...	construct P' so that AP= AP' and P' is on the perpendicular ...
and finally construct X as the intersection of AP' and the original circle.

Using the ideas above, make a script with

a. three points as given: the center of a circle, a point on the circle, and a point external to the circle; and which

b. constructs one or both points of tangency from the external point to the circle.