Mathematics Education
Last modified on June 5, 2009.
(Write-Up #7 instructions at the end . . .)
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 Geometer'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. Why?
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 contstructed 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.
1. Make script tools for construction of the tangent circles.
2. The following constructed tangent circle is not one of those in the investigation so far. Make a script tool for generating this case.
3. Discuss the loci of the centers of the tangent circles for all cases you construct.
5. Investigat your constructions of tangent circles if the two given circles intersect. (I think you will find using the script tools helpful.)
6. Discuss the locus of the centers of the constructed tangent circles when the two given circles intersect.7. Investigate, discuss, and state conjectures about the locus of the centers of the set of constructed tangent circles in Problems 5. and 6.
8. Investigate your constructions of the circles tangent to two given circles when two circular regions are disjoint. Look for all cases.
Prove that the locus of the centers, in each case, is the hyperbola with foci at the centers of the given circles.
9. Examine the trace of the tangent line in each case.
10. Consider the locus of the midpoint of the segment that formed the base of the key isosceles triangle in each construction.
11. Using animation, consider the limit of the locus of centers as the two given circles approach being tangent.
12. Is the locus ever a parabola? Is it a parabola in some limiting case?
13. Consider loci of points other than the center constructed relative to the tangent circles. For example, consider points along the line through the point of tangency to one of the given circles and the center. Or consider some points along perpendiculars to the diameter through the point of tangency (try perpendiculars not through the center as well as the perpendicular through the center).
14. Given a line and a circle with center K. Take an arbitrary point P on the circle. Construct two circles tangent to the given circle at P and tangent to the line.
Investigate. . . what else can you find?
Prepare a retrospective summary on your experience with this assignment. The summary might take a mathematical bent, stressing the underlying theorems and relationships. It might take a pedagogical bent, stressing the exploration and discovery. It might take a "here is something interesting I found" bent. Or . . . be creative . . .
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