Here is my answer to #8 from the Monge's Theorem assignment:

*I can't decide whether you would want us to teach how to
construct a tangent line to a given point on or off
the circle. For now, I am going to address the point
ON the circle *

First, I would work with the students on the construction of a perpendicular bisector of a given segment using a compass and straightedge. Then we would work on constructing a perpendicular bisector through a point on the segment. At this point, we would want to talk about the definition of a tangent line to a circle. Somehow they need to understand that the radius to the point of tangency is perpendicular to the tangent line. Geometer's sketchpad would be a useful tool for this. Once they understand this concept, they may be able to "discover" the steps to constructing a tangent to a given point on the circle and how to check that their construction is accurate.

Here are some of the sketches that I would want students to see: