The problem was posed as the following.

*Given two circles and a point on one of the circles.
Construct a circle tangent to the two circles with one point of tangency
being the designated point.*

The following is a lesson plan for a GSP exploration of tangent circles and conic sections.

Grade: 11th

Class: Pre-Calculus

Activity:

**Step 1**: Click here to see the
GSP script for the above problem.

**Step 2**: a.) Click here to see
the construction for the tangent circle to two given circles if the given
point is on the smaller of the two circles so that the smaller circle is
external to the tangent circle. Click here
to trace the locus of the center of the tangent circle(pointH). Animate
F on circle 1. What conic is formed by this trace? What point(s) are the
foci?

b.) Click here to trace the point I. What conic is formed by this trace? What point(s) are the foci?

**Step 3**: a.) Click here to see the construction for the tangent circle
to two given circles if the given point is on the smaller of the two circles
so that the smaller circle is internal to the tangent circle. Trace the
locus of the center of the tangent circle(point N). Animate F on circle
1. What conic is formed by this trace? What point(s) are the foci?

b) Click here to trace point M. Animate F on circle 1. What conic is formed by this trace? What point(s) are the foci?

**Step 4**: Construct the tangent
circles when the two given circles intersect.

a.) Click here to trace point G. What is the conic that is formed by this trace? What are the point(s) of the foci?

b.) Click here to trace point F. What is the conic that is formed by this trace? What are the point(s) of the foci?

**Step 5**: Click
here to see the construction when the tangent circle of the two given
circles intersect.

a.) Click here to trace point I. What is the conic that is formed by this trace? What are the point(s) of the foci?

b.) Click here to trace point K. What is the conic that is formed by this trace? What are the point(s) of the foci?

**Step 6: **Prepare a write-up
of your findings in your lab notebook. Please explain and describe all steps
taken during your explorations. If you did any extra investigations please
provide ask for a disk to save your scripts and GSP sketches so I may see
them. Any extra work done is worth 10 points on your grade for this lab
assignment. In order to get the full points, you must provide proofs and
pictures/scripts for me to test your results.

**Step 7**: Write a one page paper
describing one new thing that you have learned and why this would be important
to know. I.E. what occupation or place could this concept be used.

**Conclusion**: (for teacher) This
exercise is great to allow students to become familiar with using GSP. Also,
I think that it could be a great introduction to conic sections. You should
allow students to have their books on hand and let them tell you what each
of the different conic sections form. This cold be a great stepping stone
to go into many other explorations with conic sections.