For Write-up #7,
we are examining tangent circles. We will primarily examine two given circles and find the circle tangent to both of the circles. Let's start on an easier problem.
We will start with a given circle centered at A and passing through B. Find a circle tangent to this circle (we'll call it circle A) at point P that passes through point D.
The center of the tangent circle is along the line perpendicular to the circle A at point P. Thus, it is on the line AP. Further, we know the center of the tangent circle is equidistant from D and P. By definition, it is therefore on the perpendicular bisector of DP. The center is point X in the diagram below and the tangent circle is in blue.
Now we will examine when, instead of passing through D, that it is tangent to a smaller circle inside circle A. Assume we begin with a circle centered at C and passing through D that is inside circle A. Click here for diagram. We want to construct the tangent circle that is external of the smaller circle, as pictured below. The center of the tangent circle is point X.
For the steps to create the tangent circle, click here.
The smaller circle can also be in the interior of the tangent circle. A diagram follows.
Again, for the steps on how to create the circle can be found here.
One interesting exercise is to construct the center of the tangent circle (point X) and trace it as you move point P around circle A. Because the centers are different, we will get two different shapes. Click here for a GSP animation of when the smaller circle is inside and here for the smaller circle is outside.
The relationship can be examined if we trace both centers at the same time as we rotate P. That can be seen here. Note how A and C relate to the ellipse drawn from the trace points. The "trace graphs" further differ when the smaller circle is outside circle A (as here), tangent to circle A (as here), and intersect circle A (as here.)