## Lynne Bombard

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.

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.