## Tangent Circles

#### by Kent Wiginton

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.

At first I constructed a line through the point on the big circle and its center. Next I remembered that three points determine a unique circle. So I had three different points, but I realized the the circle that I wanted would not necessarily pass through the three points that I had chosen. The point on the big circle is the point that will be on the desired circle. So I decided that I would construct the segment from the point P to the center of the smaller circle. I decided that I wanted the point where the segment intersected the smaller circle to be the other point of tangency. So I constructed a new segment from point d to point p. Then I constructed the perpendicular bisector and where it intersected the original line was the center of the circle.

This worked, for a while. Everything was looking good until I moved P around the big circle and noticed that the new circle was not tangent everywhere. OOOPS!

Well I had to start over. I think I had the right idea, but I needed to fix my mistakes. After I went back and constructed the correct script for this construction things went much better.

Click here to see if the black circle is tangent to both blue circles. (check and make sure that I am right this time. ;o))

If we use the animation function on GSP we well be able to see the locus of the center of the tangent circle. (A Locus of points is all the points that the center will exist on.) We will call this center H to make things easier. There are three different places that the center could lie: inside the big circle, on the big circle, and outside the big circle. Once I began to explore with this, I learned that it is more than just where the center of the smaller circle lies it is also important where the circle exists . We will see that when the circle itself is completely outside the big circle we will see that the locus of H will be a Hyperbola. The other two places are when the center of the smaller circle lies directly on top of the bigger circle. We will have concentric circles. Moreover the locus of H will be a concentric circle as well. The last place we will look at is when a part of or all of the smaller circle lies inside the bigger on, expept when the centers are the same, we have already discussed that case. This locus will take the shape of what looks like an oval. Take a look for your self: