Investigation: 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.
For this investigation it will be necessary to look into three different 'cases.' These three cases will be where one of the circles is inside the other circle, the circles intersect, and where the circles are disjoint. In order for a constructed circle be tangent to two given circles is must be tangent to each circle at one point.
Using GSP let's first look into constructing a circle tangent to two circles where one circle lies inside the other circle. For easy understanding, I will go step by step with snapshots of my GSP file throughout this investigation of the first case.
1) Using the circle tool, create two circles one inside the other. (Make a segment defining the radius of the smaller circle with center B)
2) Draw a line through the center of the encompassing circle (center A) and one point on the circle (point C).
3) Construct a circle with center C and radius of circle B.
4) Where the new circle with center C intersects the line (outside of circle with center A) we have point D.
5) Connect circle center B to point D with a segment.
6) Find the midpoint of this segment BD and construct the perpendicular line.
7) Where this line intersects the line passing through center A and center C we will have the center point, E, of the tangent circle.
8) Construct the tangent circle with center E and radius EC.
Through these simple steps, we have created a circle tangent to two given circles, where one circle is inside another. This tangent circle is seen in purple. However, this is not the only tangent circle in existence between the circle centered A and the circle centered B. To investigate this case further, we can trace the center of the circle tangent to the two circles as the point C rotates around circle A. The tracing of the center point E is in yellow in the image below. We can see that the trace makes a clear ellipse that seems to have its foci located at the center of circle A and the center of circle B. I have also included my GSP file for your further investigation.
One Circle Within the Other
While doing the first case of this investigation, I created a GSP tool for the tangent circle when given two circles. This way, instead of having to go through multiple steps again, we can use the tool and draw out our two circles anywhere. The tool will automatically give us the circle tangent to the two given circles, as well as allow us to trace the center of the tangent circle as we have done before. Therefore, below is an image of a circle tangent to circles that are disjoint. The circle tangent to the two given circles is in purple and the trace of the center is in yellow. In case you would like to investigate more with this case, I have included the GSP file as well.
Lastly, we must investigate the case where the two circles intersect one another. Thankfully we can still use the same tool to create this case in GSP. Below is an image from my GSP file where I explored this particular case. Circles A and B intersect and we are able to see the tangent circle in purple. The trace of the center of the circle tangent to our two original intersecting circles is in yellow. As the first case, where one circle was inside the other, the trace of the center of the tangent circle created an ellipse. In addition, the two foci of the ellipse are still located at the centers of the two original circles, center A and center B. Again, the GSP file is included for your further investigation.
While this investigation seemed difficult to me at first, I found it to be extremely interesting and enjoyable! Seeing all of the mathematical ideas and technology come together is amazing.