Problem 7.1: Make a script tool for the construction of tangent circles.
See the following GSP file, if you have Geometer’s Sketchpad.
Problem 7.2: Make a script tool for tangent circles within each other.
Problem 7.3: Construct the tangent circle to 2 given circles. The given point is on the smaller of the two circles – 2 cases:
1. Smaller circle is external to the tangent circle.
First sketch shows the point on the larger circle, the second shows the point on the smaller circle.
2. Smaller circle is internal to the tangent circle.
Problem 7.4: Discuss the loci of the centers of the tangent circles for 7.1 through 7.4.
All of the circles, to this point, have loci that appear to be elliptical around the foci of each of the other tangent circles. This is shown in the constructions by noting that the sum of the distance from the center of each of the given circles to the center of the tangent circle is constant. This basically defines the focal points of the ellipse as the centers of the two given circles.
Problem 7.5: Discuss the construction of tangent circles, if the given circles intersect.
When the two circles that are given are intersecting, the tangent circle center traces a loci that appears to be elliptical around the two centers of the given circles. The main difference is that the circle reduces to a point as it crosses the intersection of the 2 given circles. The tangent circle is then interior to the second circle, and reduces to a point before it returns to the first circle.
See the following progression:
Start with the tangent circle (green) within the larger circle.
Then note that the tangent circle decrease in size as it approaches the smaller circle…
As the tangent circle reduces to a point at the intersection point and the proceeds to be interior to the smaller circle and exterior to the larger circle…
The tangent circle proceeds through the 2nd intersection between the two circles and back interior to the larger circle.
One can also check to see how the locus of the center of the tangent circle changes as the intersection of the two given circles goes from the two given tangent with the smaller interior to the larger and then when the two given circles are tangent, with the smaller being exterior to the larger:
Interior: The ellipse still remains, but is closer to being circular. The total length remains the same of the two segments between the centers as in the previous, due to the size of the circles remaining the same. The tangent circle doesn’t appear in the interior of the smaller circle, but is reduced to a point at the point of tangency.
Exterior: Again, the total length between centers remains the same. The ellipse reduces to a single line. Note that as soon as the two circles are separated, the locus of the center of the tangent circle becomes hyperbolic in shape.
Problem 7.6: Discuss the locus of the centers of the constructed tangent circles when the two given circles intersect.
Per the inclusion of the locus of the centers of the tangent circle in #5 above, and through the measurement of the segments, the segment totals remain the same, as the center moves around each of the centers of the 2 given circles. This would describe again an ellipse with foci at the centers of the two given circles.
Problem 7.7: Investigate, discuss and state conjectures about the locus of the centers of the set of constructed tangent circles in # 5 and #6.
Problem 7.8: Discuss the construction of the circles tangent to two given circles when two circular regions are disjoint.
The above sketch shows the construction of two circles with a circle tangent to both, when the original circles are disjoint. The locus of the center of the tangent circle creates a hyperbolic trace about each of the centers of the original circles.