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 weÕve 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.
WeÕll 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.