TANGENT CIRCLES

In this write up I will prepare a retrospective summary on my observations and experiences with Tangent circles.

I will make a summary that stresses the underlying theorems and relationships of the exploration.

Also I will show some mathematical and some pedagogical bent of my discovery.

This investigation begins with the following problem.

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.

We will proceed to investigate this problem and investigate
some other problems to set the direction for additional

investigations.

The Geometry's Sketchpad allows investigation, demonstration,
and exploration. It is a tool for helping develop

statements to be proved and the construction of new relationships.
The set of circles

tangent to two given circles is a very rich problem environment.
GSP helps to visualize and demonstrate; it is a means to pose
a

considerable array of related problems and investigations.

Consider the given problem. The center of the desired circle
will lie along a line from the center of the given circles with
the

specified point.

We need to find another locus for the center of the tangent
circle. Consider the problem as solved. We would have this

configuration:

Then, if we added the lines through the centers,

we would have this situation. Now consider the segment from
the center of the desired circle to the center of the second given

circle.

This segment is always of length the sum of the radius of the
desired circle plus the radius of the given circle that did not
have a

specified point. The same distance can be laid off along the line
through the given point from the center of the desired circle,
by

constructing an additional circle of the same radius with center
at the designated tangent point:

Now, an isosceles triangle is formed, like so,

and therefore the center of the desired tangent circle lies
along the perpendicular bisector of the base of this isosceles
triangle,

as follows, and now we have a construction of the desired circle.
That is, construct a line through the center of the circle with

the designated point of tangency and construct a circle of the
same radius as the second of the given circles with the designated

point as center. The intersection of the line and circle will
allow construction of the base of the isosceles triangle and hence

allow location of the center of the desired circle. The construction
follows.

Given the construction, however, consider the locus of the
center of all such circles tangent to the two given circles. With
GSP,

we can animate around the circle and trace the locus of the center
as follows:

If the center of the constructed circle is connected by segments
to the centers of the two given circles, it is immediate that
the

sum of the segments is the same as the sum of the radii of the
two given circles. This the sum is a constant and therefore the

locus of the centers of the tangent circles is an ellipse with
foci at the centers of the given circles.

The red line in the picture, that is in your construction,
is always tangent to the locus (the ellipse ).

Do a trace of the line as the tangent point of the constructed
circle moves around the large circle. An envelope of

lines is produced all tangent to the ellipse. This is essentially
the underlying technique of folding wax paper to define an

ellipse by the envelope of folds.