A tangent is the meeting of a curve or surface in a single point if a sufficiently small interval is considered (according to Webster.com).

In this investigation we will see if we can construct a circle tangent to two random circles with one point of tangency as the designated point.

We will start with two random circles. One large one, and a smaller one within the larger one. Each circle will have its center identified.

Now we will choose a random point on the perimeter of the inner circle, we will refer to this circle as Circle B.

We have labeled the random point on Circle B, i. We will now begin the construction to achieve a circle tangent to Circle A, Circle B and random point i. Now we will construct a line that runs through the center of Circle A, and this will be our base line for the investigation. We will color it black.

Now we will create a radius in Circle B from the the center of the circle to point i. We will then use this length to make a radius starting at point iii. This will create a new circle, Circle C.

We now have three circles. Circle C is the same size as Circle B, because we used the radius (orange) from Circle B to construct Circle C. Notice that point iii is now the center of Circle C and thus has under gone a name change. We now have a new point, point iv, which symbolizes the end of the radius. We will use this point, iv, to construct a new circle, Circle D. We will also create a like segment connecting Circle D and Circle B at there centers.

We will now mark the center of the line segment BD, and this new point will be v. We will use point v to construct a perpendicular line from BD. Point vi will mark the intersection of the perpendicular line and our main line that passes through the center of Circle A.

We are now ready to construct the tangent circle. We will label this circle, T, for tangent. Our new radius for Circle T, will be from point vi to the center of Circle C. (Point vi will be renamed T.)

Now we will get rid of all of our guides and just show the three circles, Circle A, Circle B and Circle T.

Now, you can use the script tool, to create your own tangent circle. To alter to the size of Circle T, play with point i, and watch the Circle T change in relationship to Circle B. Click on the script tool and open "Tangent Circle."

Tangent Circle Script Tool

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