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.
Step 1:
Start out with a large circle of any size with one arbitrary point on this circle.
Then construct a smaller circle inside this large circle and place a point on it as well.
Here is what you should have so far.
Step 2:
Here, I constructed a line through the center of the large circle to the point on the large circle.
Then, I constructed a segment from the center of the small circle to the point on the small circle.
This is what it should look like after this step.
Step 3:
For this step, I marked the radius of the small circle and then constructed a circle at the point on the large circle which would have the same radius as that of the small circle.This new circle is shown in lime green below.
Then I connected a segment from the very top point of intersection of this new circle with the line to the center of the small circle. This segment is shown in light blue. After this, I constructed the midpoint of this segment.
Below is what you should have so far!
Step 4:
The next step is to construct a perpendicular line to the segment I formed in Step 3 through the midpoint of that segment. This perpendicular line is shown in purple below.
The center of the tangent circle is at the point of intersection of the two lines.
This is what you should have after this step.
Step 5:
As I said in Step 4, the center of the tangent circle is at the point of intersection where the perpendicular line I constructed intersects with the original line. The radius of the tangent circle goes from this point to the arbitary point on the larger circle that I began with.
The final step is to select this segment that goes from the point of intersection to the point on the larger circle and construct a circle with this radius length.
Now, the tangent circle is formed. This circle is tangent to both the large circle and the small circle that I initially formed. See the tangent circle in the construction below. It is shown in dark green.
Click here to see the script tool for the construction of this tangent circle. You can click on any one of the points of the lime green circle and move it along the the large circle or you can click on the Animate Point button to see the how the circle I constructed remains tangent to the two circles at all times.
Let's investigate tangent circles some more!
There is a really cool relationship that exists concerning the center of the tangent circle. The center of the tangent circle that I was looking for lies along the perpendicular bisector of the base of an isosceles triangle. The tangent circle's center is formed at the vertex of an isosceles triangle that has two equal sides that represent the distance from:
* The center of the smaller circle we began with to the center of the tangent circle.
* The point of intersection outside the large circle of the lime green circle (the circle we constructed using the radius length of the smaller circle) to the center of the tangent circle.
As you can see below, these two sides form the equal side lengths of an isosceles triangle. These sides are represented by line segments AB and CA. Both of these lengths are equal to 2.74 cm. BC has a different length and forms the third side of the isosceles triangle with a length of 4.73 cm.
It may be useful to use this idea to check and see if you have actually found the correct tangent circle. You could just check the distances and see if two of them are the equal sides of an iscosceles triangle. How neat!
Let's look a little further to find another fascinating characteristic of the tangent circle. Let's construct a segment from the center of the tangent circle to the center of the large circle and let's do the same thing for the small circle. The centers of the large and small circles are the foci of an ellipse that exists. If you animate the arbitrary point along the large circle and trace the point that is the center of the tangent circle and also the vertex of the isosceles triangle, you will see an ellipse form right before your eyes! Click here or here if you would like some review on what an ellipse is.
Below is a sketch of the ellipse in dark red. Click here to be able to animate the point for yourself.
This concludes my investigation of tangent circles!