Tangent Circles
Assignment 7 - Diana May
An interesting problem to consider is how do you
create a circle that in tangent to two other given circles? A popular construction of such a circle
is as follows:
Start with two circles, AB and CD. Pick a point E on AB where you would
like the circle to be tangent to.
Now, construct a line through AE and mark off a
circle of radius CD centered at E.
Construct the intersection of AE and the circle centered at E and label
that point F.
Construct a line segment from C to F and label
its midpoint as G. Construct a
line perpendicular to CF that runs through G. Label the intersection of the perpendicular at G and the
line AE as the point H.
The tangent circle desired is centered at H and
has radius HE (shown in red).
As I was building this construction, I noticed
that the circle centered at E has two intersection points along the line
AE. What would happen if we chose
the other intersection and proceeded similar to before:
We still get a circle that is tangent to both
circles and is tangent at the point E.
What types of differences do we see between these two constructions?
If the two circles are disjoint, the first
construction gives us:
The tangent circle is tangent along the outside
of both circles but contains one of the circles. In the second construction (seen below), we see that it is
also tangent along the outside of both circles, but it also contains both of
the circles.
Now, what if we considered circles where one
circle contains the other?
The
first construction gives a tangent circle that is tangent along the outside of
one circle and the inside of the other.
The tangent circle does not contain either circle. In the second construction, the tangent
circle is also tangent along the outside of one circle and the inside of the
other, but it contains the inner circle.
Finally, let’s consider when the circles
intersect each other.
The
first construction gives a tangent circle that is tangent along the inside of
one circle and the outside of the other.
The tangent circle is contained in one of the circles. The second construction gives a tangent
circle that is tangent along the inside of both circles and is contained in
both circles.
To explore these two constructions for tangent
circles in GSP, click HERE.
Questions? E-mail
me.