Bill Tankersley's EMT 668 Page

Write-up 7


Tangent Circles


Problem: 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.


After exploring each of the problems presented in the assignment, I prepared a retrospective summary of my experience with the assignment. For the summary, I chose to explore every possible location of the two given circles, and the resulting locus of points of the center of the tangent circle.

First, let's look at a construction of the tangent circle, where one of the given circles (blue) lies on the interior of another of the given circles (red). Given a circle (red) with center A passing through point D, and a second circle with center B (blue) , construct a circle with center C (green) that is tangent to both circles, with the point of tangency being point D.

Click here to see a GSP file of the drawing above. Click on animate to determine the locus of point C, the center of the tangent circle. You should find that the locus is an ellipse.

Next, let's examine the case where the two circles are disjoint:

In the drawing above, the circles with centers A and B are constructed first, then the circle with center C is the tangent circle to the two existing circles. To find out what the locus of points will be for the center of the tangent circle, click here to download a sketch and click on the animate button. You should find that the locus is a hyperbola.

Next, let's examine the case where the two given circles are intersecting:

Click here to see a GSP file of the drawing above. Click on animate to determine the locus of point C. You should find that the locus is an ellipse again.

A somewhat trivial but interesting case of the tangent circles is when the two given circles are tangent to each other. In this case, the tangent circle to the two circles is one of the given circles themselves.

One of the interesting things about this assignment is trying to prove why we get an ellipse and a hyperbola for the locus of points of the center of the tangent circle. For an ellipse, we have that the sum of the distances from the foci to a point on the ellipse is always a constant. For the hyperbola, the difference to the two distances from a point on the hyperbola to the foci is a constant.

After observing some of the sketches and animations above, hopefully you can see that the foci of the ellipse and hyperbola are always the centers of the two given circles. Because the two given circles do not move, we have the fixed distances that we need for the construction of an ellipse and hyperbola.

Click here to download a GSP script for the tangent circle.



Return to Bill Tankersley's EMT 668 Page