Investigating a Tangent Circle
In this exploration, we begin by constructing a circle tangent to two given circles with one point of tangency being designated.
The following illustration shows the two given circles in the blue and the tangent circle in red. In this case, one of the given circles is completely inside the other.
Now, lets look at the tangent circle (red) when the given circles intersect.
In this illustration, one observes that the designated point has moved from the previous example. The tangent circle moves as this point moves.
Another case occurs when the two given circles are disjoint.
In this interesting case, we see that one the given circles is now in the interior of the tangent circle.
At this time in our exploration, we may wonder that the locus (purple) of the center of the tangent circle looks like. In the first case, with one of the given circles completely inside the other.
One can observe that this locus is also an ellipse and that the centers of the given circles are the foci of the ellipse.
Now we wonder what the locus is when the circles are intersecting as in our second case.
One can quickly observe that the locus is also an ellipse and the centers of two given circles are the foci of the ellipse.
Finally, we will explore the situation shown in our third case with the two given circles being disjoint. Click here to see the exploration.
In this instance, we see that the locus of the center of the tangent circle is a hyperbola with the centers of the two given circles acting as the foci.
To explore the pedalogical value of this assignment, click here.