## Assignment 7

Tangent Circles

### by Kelli Nipper

*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.*

1. Given two circles **A** and **C**, and a point **E**
on **A**.

2. Construct a line through center **A** and point **E**.
(The center of the tangent circle must lie on this line.)

3. Construct a circle with center **E** and radius equal
to circle **C**. (This radius is **EG**)

Since the center of the tangent circle must be equidistant
from point **E** and circle **C**, then if we extend the segments
by a distance equal to radius **C**, the center of the tangent circle
must also be equidistant from point **G** and center **C**.

4. Connect point **C** and point **G**. (This will
be the base of the isosceles triangle.)

5. Construct a perpendicular bisector through the midpoint
of **GC**. The third vertex of the isosceles triangle is where the two
constructed lines intersect. This point is also the center of the tangent
circle.

6. Construct a circle with center **I** and radius **IE**.

As point **E** moves around Circle **A**,
the locus of the center of the tangent circle (**I**) traces out an ellipse.
(Click on the picture to explore this using the GSP)

The foci are located at the centers of the given circles
( **A** and **C** ).--> The two sides of the isosceles triangle
( **IG** and **IC** ) are equal in length by definition. Therefore,
the distance from center **I** to center **C** plus the distance from
center **I** to center **A** is the same as the sum of the radius's
of the given circles (**A** and **C**). This is the definition of
the foci of an ellipse.

In closing, to extend this problem--Given two circles and
a point on one of the circles, investigate the tangent circle if :

- the two circles are mutually exclusive
- the smaller circle is on the larger circle.
- the smaller circle is internal to the tangent circle.
- the point is on both circles.

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