TANGENT CIRCLES

Presented by Godfried Lawson

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.

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