Let's explore circles tangent to two given circles as well as the loci of the centers of the tangent circles. We begin with two arbitrary circles and an arbitrary point X on one of the circles as shown.
The centers of the tangent circles will both lie on a line passing through the center of the circle with X on it and X itself. So we can construct this line. Additionally we construct a circle about point X with the radius of our other given circle. Do you know why?
Do you figure out why we constructed the additional circle? Consider the following: We see that the circle and our line passing through X and the center of its circle intersect twice. If we select the intersection outside of the circle, we can see that we can construct the segment from that point to the center of the smaller circle and use it as the base of an isosceles triangle by taking its midpoint and its perpindicular bisector. Then we find the bisector's intersection with the line from X through its circle's center.
The last point we found is the center of a circle tangent to both of our given circles passing through X, as the following illustration shows.
Click here for GSP tool for this tangent circle.
Consider: is this the only circle tangent to both given circles passing through X? The answer is no. Remember that the circle that we constructed about X had two intersections with the line through X and the center of its circle. Can you construct the other circle tangent to both given circles passing through X?
Click here for GSP tool for the other tangent circle.
Notice that regardless of where X falls on the circle, our constructed circles remain tangent to our given circles. Thus, by tracing the centers of our tangent circles as X moves around its circle we will obtain the loci of the centers of all circles tangent to our given circles. In the following illustration, the light red and light blue curves are the loci. Notice for this case they both are an ellipse.
Click here for GSP file with animation.
We constructed X to be a point on the larger of our two given circles. What if X had been on the smaller circle? Think about this logically. If we are constructing tangent circles, each tangent circle touches the given circles exactly once. Thus, we should be able to do the same construction regardless of the circle upon which X lies. The following illustration shows the tangent circles for two points on the smaller circle.
Just in case you are not convinced that the set of tangent circles is the same as when X was on the larger circle, the following illustration shows the loci of the centers of the tangent circles when X was on the smaller circle. Compare it to the previous loci.
Click here for GSP file with animation.
We constructed our given circles so that the smaller circle was completely inside the larger one. In this case, we observed the loci of centers for both sets of tangent circles were both an ellipse. What happens when are given circles are drawn so that they intersect? The following illustrates one set of tangent circles where the blue circle is contained in both circles, and the red circle is contained only in one of the circles.
Any conjecture on what happens to our tangent circles as the point X respective to each is moved along its circle? What will the loci look like? Examining the GSP animation file, we see that each of our tangent circles "squeezes" through the intersection points. Also notice that the red tangent circle is always external to one circle and internal to the other. The blue tangent circle is either internal to both given circles or external to both. The loci of centers for the set of "red" tangent circles is still an ellipse. However, notice that the loci of centers for the set of "blue" tangent circles is now a hyperbola.
Click here for GSP file with animation.
What about the case where our given circles are disjoint? Here we see two tangent circles for such a case.
Can you guess what the loci of centers for both sets of tangent circles are? From the illustration, we see that both are hyperbolas for this case.
Click here for GSP file with animation.
The following two illustrations again demonstrate that the location of X on one circle or the other does not effect the set of tangent circles.
Click here for GSP file with animation.
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