Janet Kaplan

Tangent Circles

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Given two circles and a
designated point on one of the circles, construct another circle that is
tangent to both circles.

WOW! Can you picture what
we are trying to do? We are going to need a great deal of visualization for
this investigation. It's a good thing we have Geometer's Sketchpad to help us
investigate, demonstrate and explore.

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Let's begin with the basic
construction. Construct a large circle (dark green) and a smaller circle (red)
inside the large one. Select a random point (E) on the larger circle. It will
become the point of tangency for the large circle. Now construct a line going
through the center of the large circle and this designated point (blue dashed
line).

Next, construct a
(temporary) circle with the designated point as the center and a radius the
same size as the radius of the smaller circle. Draw a segment from the center
of the small circle to a point on this temporary circle (brown dashed line).
Identify the midpoint of this segment and construct a perpendicular line (red
dashed line). Construct a point at the intersection of this perpendicular line
and the line drawn from the large circle's center to the designated point (B). This is the center of a circle which is tangent to both
circles.

Lastly, construct a circle
with this intersection as the center and the desginated point on the arc (light
green circle).

Notice what we have, in
effect, done. By constructing the perpendicular bisector of the segment
connecting the two small circles, we have generated the base of an isosceles
triangle. The two congruent sides of this triangle are highlighted in the
diagram below. Their length consists of the radius of the small circle plus the
radius of the tangent circle. Our construction will maintain these
relationships regardless of how we move the circles around.

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Next, we will animate point
E on the large circle and trace the center point of the tangent circle (B) to
see what pattern emerges.

To see the animation, CLICK HERE.

The locus of the center of
all circles tangent to the two given circles is an **ellipse**! An ellipse
is defined to be the set of all points such that the sum of the distances
between that point and two distinct fixed points (the foci), is a constant. In our case, the center of the light green
tangent circle is connected by segments to the centers of the two given
circles. The sum of these segments (BA + BC) is the same as the sum of the
radii of the two given circles (AD + CE). This sum is a constant, and therefore
the locus of the centers of the tangent circles is an ellipse with foci at the
centers of the two given circles (A and C).

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By altering the placement
of the line segment that connects the small and the temporary circle, and
therefore changing the intersection point of the perpendicular bisector, we can
construct a circle which is tangent to both circles, but **surrounds the
smaller circle**. CLICK HERE to see the construction and its
animation. If it is not already set, be sure to mark the intersection (B) for
tracing.

Again, the locus of the
center of all such tangent circles is an ellipse. And for the same reason as
the previous example, the foci are again the centers of the two given circles.

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Let's see what we get if we
drag the smaller circle so that it is **external** to the tangent circle.
Will a different pattern emerge, or will the same relationships hold? Think
about it first, then CLICK HERE to see. Be sure to trace point B.

Surprisingly, the same
relationship between the center of the given circles and that of the tangent
circle still hold. The sum of the segments connecting the centers of the three
circles (AB + BC) still equals the sum of the radii of the two given circles
(this time designated by (AD + 2BD) + CE). And the result is still an ellipse
with foci at the centers of the two given circles.

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One final thought to this
investigation. What will the locus of the centers of all tangent circles look
like if the two given circles intersect? It seems like a new situation. But if
there's any logic to this world, you can surmise that an ellipse will be
formed, right? CLICK HERE.

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