Tangent Circles

Rozina Essani

Given two circles and a
point I will construct a circle tangent to the two circles with one point of
tangency being the designated point. View the GSP construction for Tangent Circle.

The
construction was made by selecting a point on the outer circle and making a
circle equal to the smaller circle
around that point. Then I connected the center of the inner circle to the point
on the similar circle that lies on the diameter of the large circle. Then I
found the perpendicular bisector of this segment and the point where the
bisector intersects the diameter of the larger circle. This intersection points
now becomes the center of the tangent circle and radius of this circle is the
center to the point on the larger circle.

Another case is when
the tangent circle engulfs the inner circle. View the GSP construction here.

In this case we find
the perpendicular bisector of the segment with endpoints center of inner circle
and the point, on the similar circle that was constructed, that lies on the
diameter of the larger circle on the inside of the larger circle. This gives us
a tangent circle that fully engulfs the inner circle.

Now let us look at the loci
of the centers of the tangent circles for both of the above
mentioned cases.

Case 1:

This allows us to see
that the loci of the center of the tangent circle is
an ellipse with foci at the centers of the two circles, the inner and the outer
circle.

Case 2:

We can see from this
construction that the loci of the center of the tangent
circle in this case is also and ellipse. This time the ellipse is within
the inner circle and its foci are still the centers of the inner and outer
circles.

Now lets look at the
tangent circles when the two given circles intersect.

Case 1:

Looking at the path of
the tangent circle we can see that once tangent circle reaches the point where
the circle is outside the outer circle the tangent circle changes its path to
being tangent on the inside of the inner circle and outside of the larger
circle simultaneously.

The loci
of the tangent circle is this case is still an ellipse with foci as the
two centers of the given circles.

Case 2:

In case 2 the path of
tangent circle is the exact opposite as in case 1. The tangent circle changes
paths in the part where the smaller circle is inside the larger circle.
Interestingly, the loci is no longer an ellipse in
this case. It is now parabolic keeping the foci as the centers of the two given
circles.

What if the two given
circles are disjoint?

Case 1:

When the two given
circles are disjoint the tangent circles is tangent to
both circles simultaneously on the outside. The loci of the
center of the tangent circle in this case is also parabolic with foci
the center of the two given circles.

Case 2:

Again in this case the
tangent circle is tangent to both given circles in the area outside both
circles. The loci is a more stretched out parabola
with foci staying as the centers of the two given circles.

Tangent Line traces:

Case 1:

Case 2:

The tangent line traces
coincide with the trace of the loci of the tangent circles for each case.

Lets see how the locus
of the midpoint of the segment that formed the base of the key isosceles
triangle behaves.

Case 1:

Case 2:

When I traced the
midpoint of the base of the isosceles I noticed that it creates a circle and no
matter where we place the smaller circle, the trace circle remains the same
size in both cases.