By:† Lauren Lee
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.
To construct a circle tangent to the two circles above, I used GSP.
The construction is as follows:
First I chose an arbitrary point on the large circle and construct a line that goes through both the point and the center of the circle.
Next I construct the radius of the small circle.† I use this radius to form a new circle.† My new circle is constructed using my arbitrary point as the center and the radius of the small circle.† I also construct a segment between the far intersection of the new circle and the line that goes through the center of my large circle.
Next I construct a segment from the midpoint of the small circle to the far point of the new circle.† After I have done this, I find the midpoint of the segment.† Next I use the midpoint of the segment and the segment itself to construct a perpendicular line.† My perpendicular line intersects the line through the center of my circle.† This point of intersection is the center of my tangent circle.
Now the last step is to construct the tangent circle using its center and my arbitrary point on the big circle.
Now that I know how to do the construction, I will investigate the locus of the tangent circle.† With GSP, I trace the locus of the tangent circle while my arbitrary point moves around the large green circle.
The red circle is the traced line of the movement of the locus.† It appears to be an ellipse.† Notice that the orange perpendicular line is tangent to the ellipse.† Will it always be tangent as our arbitrary point moves around the circle?
Yes, we can see that the line will always be tangent to the ellipse.
Now letís investigate a second tangent circle of the two green circles.
This tangent circle will be constructed in a similar way.† The difference is that instead of constructing a segment from the midpoint of the small green circle to the far point of intersection of the blue circle and the line through the center, I will construct a segment to the lower point of intersection of the blue circle and the center line.
The rest of the construction is the same, resulting in the following red tangent circle.
Now we have a case where one of the green circles is inside of the tangent circle.† Letís see what happens when we trace the locus of the tangent circle.
As you can see, the locus of the tangent circle appears to form an ellipse.
Now letís investigate the tangent circle of two intersecting circles
The tangent circle is as follows:
What about the locus of the tangent circle?
By tracing the locus, we can see that the ellipse goes outside of the large circle.
What about the perpendicular line?† Will it still always be perpendicular to the ellipse?† YES!
Letís do another investigation:
What happens now if we change the construction as before to find the other tangent circle?
Letís construct a segment from the midpoint of the small green circle to the lower point of intersection of the blue circle to find the other tangent circle.
Now both of the green intersecting circles are inside of the purple tangent circle.† What will happen with the locus of the tangent circle?
You can see that the locus of the tangent circle has formed a hyperbola.† So what will happen when we trace the perpendicular line?
The perpendicular line is still tangent to the locus of the tangent circle.
From these investigations, we can see that the locus of the tangent circle does not always form an ellipse.† Also, the perpendicular line will always be tangent to the locus of the tangent circle.
If you want to investigate further, here is my script tool for