Assignment 7

Tangent Circles

by
Jenny Johnson

What are tangent circles?

Two circles are tangent if they only touch in one point. A circle is tangent to two circles if it touches each of those circles in only one point each.

How do construct a circle that is tangent to two other circles?

Let us start with two circles – one inside the other – and construct a circle that is tangent to each of the two circles. The picture below shows the two given circles and the point on the larger circle that will be the point of tangency.

In order to draw the circle that will be tangent to both circles, let us connect the center of the larger circle with the point of tangency we’ve chosen with a line. Then construct a circle congruent to the small circle with center at point A, as shown below.

Now we will construct a line from the center of the smaller circle to the point of intersection (B) of the new circle and our line. Then we find the midpoint of the segment from the center of the small circle to B.

We can then construct the perpendicular through the midpoint. This perpendicular line intersects the original line through the center of the larger circle. The point at which they intersect will then be the center of the circle that is tangent to both circles.

Now if we animate the point A, we can watch how the circle we constructed changes as the point of tangency moves on the outside circle. We can trace the center of this circle as point A moves around the circle. The locus of the center of this tangent circle will then look like an ELLIPSE as seen in the picture below.

It also appears that the foci of the ellipse are the centers of both original circles. To see the animation, click here. Now let us look at the sum of the distance from the center of the small circle to the center of the tangent circle and the distance from the large circle to the center of the tangent circle. When we animate point A, the sum of these two distances remains constant. Thus, we can feel confident that the two centers are the foci of the ellipse. To see the animation, click here.

Are there any other circles that could be tangent to both circles at the same time?

It does seem that there could be another circle that would be tangent to both circles. I constructed the line from E to A and marked the point where that line intersects the circle centered at A. We’ll call this intersection F. Then construct the perpendicular bisector of CF. The point where the perpendicular bisector intersects line EA is the center of another tangent circle. The new tangent circle is purple.

What will happen if I trace the center of the new purple tangent circle?

Will the traces form another ellipse? I traced the center of the new circle (G) while the point A moved around the big circle. The locus of the center of the circles was another ellipse.

To explore this locus, click here. It appears that the centers of the given circles are the foci of this ellipse also. If we measure the sum of the distance from C to G and the distance from E to G, we see that the value is constant. Just as with the other ellipse, C and E are the foci of the ellipse. To see an animation of these distances remaining constant, click here.

What if we trace the centers of both tangent circles?

We can gather that the loci of the centers of the two tangent circles are both ellipses with the same foci C and E.

What happens if we move the smaller circle so it overlaps the bigger circle?

If we move circle C so it overlaps circle E, then the loci of the centers of our tangent circles are a little different. The locus of point D is still an ellipse with foci C and E, but the locus of point G becomes a hyperbola with foci C and E. See the picture below. Click here for the animation.

What happens to the tangent circles if circle C is moved completely outside of circle E?

When we move circle C outside of circle E, the centers of the tangent circles both form hyperbolas when traced. For a picture of the loci, see below. For an animation, click here.

Thus, we see that for any two circles, there will always be two circles tangent to both circles. Depending on the position of the original two circles, the loci of the centers of the tangent circles will form either ellipses or hyperbolas with the foci being the centers of the original two circles.

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