### The Construction of the Circle Tangent to Two Given
Circles

#### by: BJ Jackson

This construction starts with two given circles. It does not
matter if one circle is inside the other, or if the circles intersect,
or if the circles are disjoint. The construction will work for
all three cases. To me, it is easier to see the construction for
the situation when on circle is in the interior of the other and
that is what the construction that will be presented here. This
is done because GSP allows the user to move the cirlces into any
position and the construction still holds. Note: if you want to
make a tool in GSP, the given circles should be constructed from
two points not from the circle tool.

So, start with two circles and an arbitrary point(A) on one
of the circles.

Next, construct the line through the center of the the large
circle and A. Also, constuct the radius of the small circle. Now
the construction should look like this.

We now need to constuct another circle with radius equal to
the length of the small circle with its center at point A and
find the point B where this new circle intersects the line.

Next, constuct the segement from point B to the center of the
small circle. This gives us the base of the isosceles triangle
that the construction is dependent upon. While we are at it, find
the midpoint of the segment and the line perpendicular to the
segment at the midpoint.

The last two steps of the constuction are two find the point
where the two lines intersect which is the center of the circle
tangent to the two given circles. Now, use the newly found point
and point A to constuct a circle and this circle will be tangent
to the two given circles.

Click **here** for a GSP
sketch and tool of the above construction.

Another interesting construction is the circle tangent to the
given circle on the interior and the tangent circle. Click **here** for a GSP sketch and tool
of the constuction. A picture of the constuction is below.

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