Christa Marie Nathe
Presented with the problem where we are given two circles and a point on one of the circles, and we want to construct a circle tangent to the two circles with one point of tangency being the chosen point. The figure below will be the construction we will use to derive the tangent circle.
Specifically what our objectives are is to find one common point to the outside of the small circle and one common point to the interior of the large circle that will be yield the radius of the desired tangent circle.
In order to proceed in creating the desired tangent circle, we must first acknowledge that the center of the preferred tangent circle will be located on a line going through the center of the large circle.
By marking the radius of the small circle we can make an identical circle at the apex of the large circle where the center point is on the large circle, as shown below.
The next step is to connect the center point of the small internal circle to the top of the small dashed circle and plot its midpoint.
The objective in connecting those two points and finding the midpoint of the segment is to help us determine where the center point of the desired tangent circle will be located. By drawing a perpendicular bisector of this fragment we can determine the center point for our tangent circle. The point of intersection of the vertical line through the center and the perpendicular bisector will that point.
As we can see, the red circle is tangent to one point on the exterior of the small circle and tangent to one point on interior of the large circle.
Do not be deceived by this construction. We can manipulate our figure and the red tangent circle will still maintain its properties.
Observe the following maneuvers to see for yourself.
Now lets take a look at the original construction, and trace the tangent line that intersects the center point of the tangent circle.
As we can see an ellipse is created in the wake of tracing the tangent line.
What we did here is to mark the loci of the tangent center point as the tangent point is moved along the perimeter of the large circle. An ellipse is a collection of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.