Circles that Touch

by Zack Kroll

We are given two circles and a point on one of the circles. From there we construct another circle that is tangent to the two original with one point of tangency being the designated point.

Here is a script tool that can be used in order to construct tangent circles.

We begin with a circle located inside of another with a point chosen on the edge of the larger circle. From there we are able to construct another circle through that point that is tangent to both of the original circles.

From there we decide to investigate the locus of the points that are created by the two tangent circles. There are three cases that we will look at and discuss.

The first case looks at what occurs when one circle is located inside the other. The locus that is formed in this case is two concentric ellipses. As the inner circle gets smaller, the two ellipse stretch more horizontally. As the inner circle becomes larger, the ellipses appear to look more like circles.

What would happen if the center of the two circles was the same point, but one was still smaller than the other? When the center of the two circles merge to become the same point, the ellipses become circles as well creating four concentric circles.

In the second case the smaller of the two circles is located partially outside of the larger one. The locus in this case is two different conic figures, an ellipse and a hyperbola.

Finally in the third case the two original circles are disjoint (not intersecting and not inside of one another). The resulting construction is two hyperbolas. As the outer circles moves further away from the larger one, the hyperbolas experience a vertical stretch.