Assignment 7:
Tangent Circles
by
Ángel M. Carreras Jusino
Goal:
- Make GSP constructions and script tools for construction of the tangent circle.
- Investigate and state conjectures about the loci of the centers of the tangent circles when:
- One given circle lies completely inside the other
- The two given circles overlap
- The two given circles are disjoint
This script will ask you to construct two circles, after that the script will automatically will create a point P on the first circle.
The script will construct the two circles that are tangent to the circles that you have constructed at the point P.
After your construction is completed it should look like the following applet.
This applet and the script tool enable you to investigate:
- When one circle lies completely inside the other.
- When the two circles overlap.
- When the two circles are disjoint.
Observations and Conjectures
When one circle lies completely inside the other, the locus are:
An ellipse that has as focus the centers of the two circles and the sum of the distances from any point in the ellipse to the focuses is equal to the difference of the radiuses of the two circles. If the two circles are concentric then the locus is also a circle
An ellipse that has as focus the centers of the two circles and the sum of the distances from any point in the ellipse to the focuses is equal to the sum of the radiuses of the two circles.
When the two circles overlap, the locus are:
A hyperbola that has as focus the centers of the circles and the difference of the distances from any point in the hyperbola to the focus is equal to the difference of the radiuses of the two circles.
An ellipse that has as focus the centers of the two circles and the sum of the distances from any point in the ellipse to the focuses is equal to the sum of the radiuses of the two circles.
When the two given circles are disjoint, the locus are:
A hyperbola that has as focus the centers of the circles and the difference of the distances from any point in the hyperbola to the focus is equal to the sum of the radiuses of the two circles.
A hyperbola that has as focus the centers of the circles and the difference of the distances from any point in the hyperbola to the focus is equal to the difference of the radiuses of the two circles.
From the applets shown above is clear that the loci of the centers of the tangent circles can be ellipses, circles, and hyperbolas. It would be interesting to investigate if there is any case in which the loci of the centers are parabolas.