University of Georgia
*see
other examples
Tangent Circles:
We begin with two circles, one
inside of the other. Now, we select a point on the smaller circle
and we can construct a circle tangent to the small circle, at
that point, and tangent to the larger circle. In fact, we can
construct 2 such circles. One of these circles is disjoint with
the area of the small circle (except at the tangent point) and
the other contains the smaller circle.
Here are a couple of scripts that produce the descibed
circles:
Internal Tangent Circle
External
Tangent Circle
Now, let's look at a couple of examples:
The picture shown above
displays the internal tangent circle (in red) and the locus of
its center (in green) for the 2 blue circles, as the point of
tangency on the small circle moves around its circumference.
The circle shown in red is the
external tangent circle for the 2 blue circles. The green ellipse
is its locus as the point of tangency on the small circle moves
around its circumference.
*We can show that the two green paths are,
indeed, ellipses. What's more, we can show that they share the
same focal points, which are the centers of the blue circles.
Let the following graph illustrate:
This is a graph of the 2 tangent
circles (in green and red) to the 2 blue circles shown. The locus
of the centers of these circles, as the point of tangency to the
small circles changes, are shown by the 2 ellipses.
Take a moment to
explore the construction displayed above by clicking on the picture.
Here is something interesting...
1) Focal Points: We
can note that the focal points of the locus of centers of the
2 tangent circles are given by the light blue points shown above.
These points are also the centers of the 2 original (blue) circles.
This observation is evident bacause the sum of the lengths of
the green segments and the sum of the lengths of the red segments
remain constant as the green and red points trace the ellipse.
This means that the green segments are the focal radii for the
small ellipse. Likewise, the red segments are the focal radii
for the larger ellipse.
2) Colinear points: Also, the centers of the 2 tangent
circles, the center of the small blue circle and its point of
tangency all lie on the same line or segment. For the purpose
of this discussion, let's call this the Norton segment. Also,
the Norton Segment, with the center of the big blue circle, forms
the Norton Triangle. Let's look a little closer at the Norton
Triangle...
3) Focal Radii: The focal radii of the
larger ellipse are measured and added at the top of the display.
The focal radii sum of the smaller ellipse is at the bottom. Note
that the segment AE is the radius of our smaller circle. Twice
the length of AE is the difference in length of our focal radii
sums. Why?
To see this we can look at a special case --
where we have an isosceles triangle...
Before we return to our explaination of focal
radii, let's make one last observation, en route.
4) Tangent Circle Radii: When the length of segment CH
is equal to the segment CS, we have tangent circles of equal size.
This is because C is the center of the big circle, and both tangent
circles, inside of the big circle, are tangent to the big circle.
Their radii, the difference of the big circle's radius length
and the length of CH/CS, are equal.
Back to focal radii: The radii of these tangent circles,
HE and SE, bisect the 3rd side of the Norton Triangle at point
E. Now, remember the focal radii of the two ellipses are displayed
in red and green. Let's turn our focus to the segment AE. If AE
were green, we would have the sum of the focal radii equal for
both ellipses. Instead it is red, so that length has been subtracted
from the green sum and added to the red sum. Therefore, our perevious
statement about the difference of sums of focal radii is justifyable.
For now, this concludes our study of tangent circles and the Norton
Triangle. Feel free to explore in GSP or...