By: Melissa Wilson

In this investigation we will look at creating a circle that is tangent to two other circles, which have varying relationships to each other.

Case One:

The two circle are disjoint, which means they do not intersect. We look at circle A and circle B, shown below.

We will select the point on circle B that will be our point of tangency. Let's call this point C. Now we will create a line through point B, which is the center of circle B, and point C. Also we will create a circle with a center at C but with a radius equal to the radius of circle A. This is shown below.

Line BC intersects the circle we created at C in two places. We will first look at the point outside of circle B. labeled D (we will use the point inside later). We create a segment from this point to point A, which is the center of circle A. Now we find the midpoint of this segment, labeled E. Construct a perpendicular bisector of AD through E. This will intersect line BC at point F. This is shown below.

We know that the distance of AF and of DF are equal since F lies along the perpendicular bisector of AD. We also know that the radii of circle A and circle C are the same by construction. Therefore we know that the distance from F to points on both circle (such as point C on circle B) are the same. We will draw the circle centered at F with a radius of FC, as shown below. The new circle (shown in light purple) is tangent to circle B and circle A.

Without our construction shown:

There is another circle that can be constructed that will tangent to circle A and circle B. This circle will be done using the other point from where circle C intersected line BC, labeled P. The construction is the same as before. The midpoint of AP is Q and the perpendicular line through Q intersects line BC at point R. The tangent circle here is centered at R with a radius of RC.

Now without our construction shown:

Here is both tangent circles shown:

Case Two:

The two circles are intersecting. The construction for both tangent circles remains the same. Below you can see the construction for the tangent circles for when circle A and circle B are intersecting.

Now without the construction shown:

Case Three:

One circle contained inside the other. As before, the construction for these tangent circles remains the same for when circle A is completely contained inside of circle B. The construction is shown below.

Now without the construction shown:

The GSP file that was used to create this assignment can be found here. Use it to look at various arrangements of circle A and circle B.