**Using
GSP to explore Tangent Circles**

This investigation centers around
constructing a circle tangent to two other circles. We can focus on how
technology (specifically in this case GSP) aids and encourages exploration.

While this construction is not extremely
difficult, it does require several steps. Considering different shapes and
configurations of circles could take a great deal of time just to construct
separate examples. A student is likely to get bogged down in the steps of the
construction and miss the point of the exploration if they are required to
start over from scratch for each idea they might have.

Fortunately, programs like GSP make the
construction work relatively short and painless so that the user can
concentrate on the exploring their construction. For example, there are many
cases for constructing a circle tangent to two circles. The two circles may be
disjoint. They might intersect each other. One might even be contained in the
other circle. These cases are illustrated below (initial circles in red and
tangent circle in blue).

Doing each of these constructions from
scratch would be time consuming (and frustrating). GSP, however, allows the
user to take the first construction and change the shapes of the two initial
circles until they intersect while keeping the tangent circle construction
intact. Thus we arrive at the second construction in a matter of seconds. Then
to arrive at the third, we just need to move one of the initial circles inside
the other.

We can then either save the initial
construction or make it into a script tool. Then the user can quickly begin
exploring new ideas without having to first take the time to recreate the
construction.

Some interesting things to explore in this construction are the loci of the center of the tangent circle. This can quickly and easily be done in GSP by tracing the tangent circle center and animating the arbitrary point on the larger circle used to construct the tangent circle. We show an example of this below (locus in green).

We can then quickly observe that in this case
the locus is an ellipse that appears to have foci at the centers of the initial
circles. We can then quickly see what the locus is when the initial circles are
dis-joint.

Here we can see the locus is a hyperbola
again seeming to have foci at the centers of the initial circles.

Using a piece of technology like GSP
encourages creative thinking and a curiosity to explore new ideas by limiting
the amount of time the student spends to mechanically construct those ideas. It
diminishes the chance that a student will get frustrated at exploring an idea
that ultimately does not work out and then give up on trying another idea they
may have.