Tangent Circles

I will
investigate a circle tangent to two given circles.

First,
letŐs start with what we are given: two circles, c1 and c2, one inside the other with centers
A and B.

Next, I
will place a sliding point on each circle. LetŐs label those points C and D.

Now, I will
construct a line segment from B to D, which is the radius of radius c2. I will also create a line going through points A and D.

Next, I
will construct a circle with a center at the sliding point C, with radius BD.
LetŐs name this circle c3.

Now, I will
construct the intersection of circle c3
and line AD, on the exterior of circle c1.
LetŐs label this intersection point, E.

Now, I will
draw a line segment BE and construct its midpoint M.

Draw a perpendicular
line through BE at M. The intersection of this perpendicular line and line AD,
will be named F.

Now, I can
construct another circle, c4, with center at F and radius FC. This new circle
is the circle that is tangent to the two given circles.

Now, if I
hide all of the additional constructions, I can see the tangent circles more
clearly.

What will
happen if I slide point C around the circle? If I animate point C, then some
intersecting things happen.

First, the
circle always remain tangent, but the radii of the circle centered at F changes
as C moves around the circle.

Also, letŐs
trace F as D is animated. I see that point F moves in an ellipse with foci A
and B.

Click **here** to animate.

Wow, isnŐt
that AMAZING!!!!!!!