Case
4: Purple Circle:The
given point is on the small circle, and the small circle is internally tangent
to the constructed circle when the small circle is in the interior of the
big circle.

This
picture shows the construction of the tangent circles when the given point
is on the large circle. The
blue circles are the givens.
The pink circle is externally tangent to
both given circles, and the green circle is internally tangent to the big
circle and externally tangent to the small circle.
Click here for a sketch with the construction
lines left in. To construct
draw a line through the given point and the center of the circle on which
the given point lies (call this one circle A, and the other given circle
B). This is the aqua line in
this picture. At the given point,
construct a circle with circle B's radius (call this one circle C). To
get the pink circle, construct a segment from the 'far' intersection of
circle C with the line to circle B's center. To
get the green circle, draw a segment from the 'near' intersection of circle
C with the line to circle B's center. These
are shown as purple segments in the sketch. In
either case, get the midpoint of the segment and construct a perpendicular
at that point. The perpendiculars
are shown in orange. The intersection
of the perpendicular with the original line is the center of the tangent
circle.
If
the given point is on the small circle, interchange the circle A's and
circle B's in the description above, and the construction holds. For
this scenario, the red circle mimics the green circle above, and the purple
mimics the pink circle. Click
here to see the construction
The
center of the tangent circle traces out some interesting patterns.
How do these patterns compare among the different 'flavors' of tangent
circles described here?
When
the two given circles are disjoint or intersect, the center of the tangent
circle traces a hyperbola. As
the two blue circles get closer together, the hyperbola flattens out. This
flattening continues until the small circle is almost inside the big circle:
When
the small circle is close to being internally tangent, the pink circle
'disappears' (it's actually coincident with the big circle) except for
an instant as the point is rotated around the big circle. The
center of the tangent circle traces out an ellipse so flat that it looks
like a line.
When
the given circles are disjoint, the center traces out the hyperbola. This
looks similar to the case with the pink circle.
When
the two given circles are externally tangent,
the center of the tangent circle traces out the flat ellipse seen here.
When
the given circles intersect, the center of the tangent circle traces out
an ellipse, not the hyperbola seen with the pink circle.
The
green circle also traces out this ellipse when the small given circle in
on the interior of the big circle.
In
this scenario, the two blue circles are the given circles, and the given
point is on the small circle. One
of the blue circles is always internally tangent to the red circle, and
one is always externally tangent. Because
of this, we should expect the traces to look the same as for the green
circle above. Click
here for this sketch.
When
the circles are disjoint, the familiar hyperbola is traced out. The
hyperbola gets flatter as the centers of the two given circles approach
each other.
When
the two blue circles are externally tangent,
the flat ellipse is traced out.
As
with the green circle, when the two circles intersect or when the small
circle is on the interior of the big circle, an ellipse is traced out.
Here's
the purple circle. Check out
the sketch to make sure that you can validate
that the purple circle center traces out the same patterns that the pink
circle's center did.
where
d(P, F1) is the distance between points P and F1, d(P, F2) is the distance
between points P and F2andd(F1,
F2) is the distance between F1 and F2.
F1
and F2 are the centers of the given circles and the foci of the hyperbola. Since
only the tangent circles and their centers vary, and not the given circles
or the distance between them, it's easy to see that the figure is always
a hyperbola.