This investigation begins with the following problem.
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.
Below is my investations into tangent circles.
Script #1: Constructs a circle that is tangent to the given circles.
Figure 1: The smaller given circle is external to the contructed tangent circle.
Script #2: Also constructs a circle that is tangent to the given circles.
Figure 2: The smaller given circle is internal to the contructed tangent circle.
After constructing the tangent circle to two circles it is interesting to see the locus of centers of the tangent circle.
See Figure 3: Circle AB and circle CD are given. Point P is the given point of tangency for circle AB.
Circle EP is the tangent circle. As the given point P changes, the tangent circle changes.
Figure 3: The centers of all tangent circles for the two given circles is represented below by the ellipse.
The construction of the tangent circles when the two given circles intersect is also interesting.
See Figure 4: one of the tangent circles is internal to one of the given circles, and the other
tangent circle is external to both given circles.
Figure 4: Circles IP and KP are tangent to the given circles.
Another interesting aspect of these tangent circles occurs when looking at the loci of the centers.As anticipated, one of the loci is an ellipse (in Figure 4, for the tangent circle IP).
The other locus (in Figure 4, for tangent circle KP) is a hyperbola.
Linked below to GSP shows a demonstration of the two loci.
Click Here! to see the ellipse and the hyperbola.
This investigation has allowed me to make the following conjectures.
Given two concentric circles, the loci of the centers tangent circles are both circles.
Given two circles (one interior to the other), the loci of the centers tangent circles are both ellipses.
Given two tangent circles (one interior), there will be at most one tangent circle whose
locus of the centers is an ellipse.
Given two tangent circles (one exterior), there will be at most one tangent circle whose
locus of the centers is a hyperbola.
Given two circles intersecting in two points, one locus is an ellipse and the other is a hyperbola.
Given two circles (mutually exclusive), the loci of the centers of the tangent circles are both hyperbolas.
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