Tangent Circles

(IÕll try to not go off on a tangentÉ)

By: Carly Cantrell

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.

The two circle can either be one inside of the other,
overlapping, or completely disjoint.

Use my script tool to see for yourself!

If you were constructing on your own you would want to
do the following:

Create any two circles. On one of the circles place a
point and make a line through the center of the circle and this point. Next,
select the radius of the second circle and the point you just created. Then
ÒConstructˆ Circle by Center + RadiusÓ and now you will make a
point where this circle and the line you constructed intersect outside of the
circle. Next, make a segment from the center of the second circle to the point
you just constructed. Find the midpoint of that segment. Then create the
perpendicular bisector of that segment. Make a point of the intersection of the
perpendicular bisector and the first line you constructed. This point is the
center of your circle of tangency! Create a circle from that point to the very
first point you constructed on the first circle.

LetÕs compare the tangent circle when the two circles
are inside one another, overlapping, and disjoint:

LetÕs investigate the loci of the centers of the
tangent circles for all three cases:

When the smaller circle is within the other you get
the following loci:

The locus in this case is an ellipse.

We know this is an ellipse because the centers of the
two green circles are the foci of an ellipse.

When the circles are overlapping you get the following
loci:

The locus in this case is an ellipse.

Again, this is an ellipse because the centers of the
two green circles are the focal points of the locus ellipse.

When the circles are disjoint you get the following
loci:

The locus in this case is a hyperbola.

This case is different because when the circles are
disjoint the centers of the two circles each act as the foci of a hyperbola.