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Assignment #7
An Investigation of Tangent Circles
By Kevin Mylod

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Most of know what a circle is and its properties, but what exactly is meant by the word, tangent? By definition, tangent means the intersection of two objects (in this case two circles) at one distinct point. You can see by the illustration above that the two given circles (in blue) are both tangent to the red circle at one distinct point (c1 at point E and c2 at point J).

It is also interesting to see what happens when we move these two given circles around. Consider the locus of the centers of all such circles tangent to the two given circles as one such investigation. Recall that the term locus refers to the collection of points that satisfy certain given conditions. In this case, our locus of points refers to the location of all the center points of the circles of tangent of the two given circles if we were to move the given point E around its circle, c1.

To see how the above image was constructed, check out the GSP script and follow the instructions in the "comments" section.

In the following Examples the Locus of Points of the centers of the tangent circle (in red) to the two given circles (in blue) will always be illustrated in Green. Notice that the given point will always be on the larger of the two given circles. This is done for clarity-sake. It is certainly not a rule for this assignment.

`One given circle is internal to the other `

It very much resembles a circle, and appears to be a circle when the centers of the two given circles are coincident.

`Both given circles Intersect each other`

Notice that the intersection of the two blue circles gives the locus of points a more elliptical display.

`No Intersection between the two given circles `

The locus of points is now in the shape of a hyperbola

To investigate the changing shape of the locus of points yourself, go to the GSP sketch.

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