Brenda King

Retrospective summary of

Tangent Circles

__Introduction:__

This investigation had to do
with the following problem:

Given two circles and a
point on one of the circles, construct a circle tangent to the two circles with
one point of tangency being the designated point.

For the purpose of this
write up, I distinguish between the two given circles as large and small. The __constructed__ tangent circle will be colored red in
all diagrams and the __given__ circles will be colored black. The green line segments will be used to determine
relationships that were discovered from the trace and locus of points in the
investigation.

While doing this
investigation, I explored six cases of tangent circles. The cases depended on two things 1) the
placement of the given circles (inside, intersecting or outside each other) and
2) the construction of the tangent circle either around or exterior to the
smaller circle. The following diagram
shows the six cases considered.

__Trace and locus resulting in
an ELLIPSE__

An ellipse is the set of all points, C(x,y), the sum
of whose distances from two distinct points (foci), S(x,y) and L(x,y) is
constant.

The following 3 traces and locus arrangements resulted
in an ellipse. The trace of the midpoint was a circle.

The diagram below identifies key components for
discussion.

Center of small
circle, S. Radius small circle,
r.

Center of large
circle, L. Radius of large
circle, R = segment PL

Center of tangent
circle, C. Radius of tangent
circle, R_{T} , segment PC

See green segments
for the following discussion.

For Case #1 and #2

Length of CL = PL
– PC = R - PC

Length of CS = PC
+ r

Sum of CL + CS = R
– PC + PC + r = R + r = Radius large + Radius small = constant

For case #4, where tangent circle is outside small given circle,

Length of CS = R -
PC

Length of CL = PC
– r

Sum of CS + CL = R
– PC + PC - r = R - r = = Radius large - Radius small = constant

__Trace and locus resulting in
a Hyperbola__

A hyperbola is the set of all points C(x,y) the
difference of whose distances from two distinct fixed points (foci) is a
positive constant.

The case #3, 5, and 6 resulted in a hyperbola. The midpoint again was a circle in all cases.

The locus of points alone are shown below:

I found it interesting that in case 3, the **locus of
points ** displayed a parabola where the **trace of the tangent line** showed
a hyperbola. This result must be from
a limiting situation special to case 3.

Case #5 and #6 are disjoint circles (small circle
completely outside large circle). Both
resulted in a hyperbola trace and locus.

Interestingly, where the two given circles intersect
each other, case #2 and #5, the results were different from each other. For case #2, an ellipse pattern was found and
in case #5 a hyperbola was found. If I
did further explorations, I might be able to discover why Case #4 resulted in
an ellipse, even thought the construction of the tangent circle was like both
case #5 and #6.

__Summary chart__

Here
is a sketchpad file
with all the results pulled together.