Exploring Tangent Circles Using GSP
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.
I chose to create a script tool and a gsp animation file for the tangent circle problem. The procedure below briefly outlines my constructions. My gsp files are available at the end of this page. You must have Geometer's Sketch Pad to open the files.
Construct two circles of different sizes. The smaller circle should be inside the larger circle. Construct an arbitrary point on the large circle.
Construct an arbitrary point on the small circle and then construct a radius to the point.
Construct a diameter of the larger circle, passing through the arbitrary point constructed in Step 1. Make sure it is a diameter LINE. I put a segment here to illustrate placement of the LINE, but make sure you continue the line in both directions for the animation to work.
Construct a third circle with the arbitrary point of the large circle as the center, and the radius the same length as the other constructed radius of the purple circle.
Mark the point of intersection of the orange diameter with the newly created circle. Use the intersection point which is outside of the large green circle.
Construct a segment from the new point to the center of the original small circle, and marke the midpoint.
Construct a perpendiular line through the same segment through the midpoint. Mark the point of intersection with the diameter as P.
Finally, construct the new circle with center P and the point on the circle being the center of the third constructed circle.
Make sure that if you have read this far, you open up the gsp animation file to see the tangent circles in motion. It is pretty neat. If you would like to use my tool, this gsp file allows you to easily create your own tangent circles.