**Assignment 7**

**This investigation begins 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.**

I chose to write up problem number four from this assignment:: **Discuss
the loci of the centers of the tangent circles for all case you construct.**

Given two circles and a point **P** on the first
circle, there are actually two different circles tangent to the first circle
at **P** and tangent to the second circle:

For two circles to be tangent at a point **P**, their
tangent lines at **P** must conicide. From basic Euclidean geometry,
we know that the tangent line at the point **P** of the circle
centered at **C1** is perpendicular to the segment **PC1**.
Thus any other circle tangent to **C1** at ** P **must
be centered on the line through **P** and **C** (colored
red in the folowing diagram):

For this circle to also be tangent to the circle centered at **C2**,
it must also be centered at a point ** Q** on the red line where
**QP** is congruent to ** QC2**.

**Circle 1**

If we translate the circle centered at **C2** to
be centered at **P**, then it intersects the red line at a point
**R** outside the circle centered at **C1**. Any point
on the perpendicular bisector of **RC2** (shown in red) is the
same distance from **C2** as it is from ** R**. At
the intersection **Q** of the two red lines, since the circle centered
at **C2** and the circle centered at ** P** have the
same radius, **QS** is congruent to **QP**, and since
**C2**, **S**, and ** Q** all are on
the same line, the circle centered at **Q** and the circle centered
at **C2** are both tangent to this circle.

**Circle 2**

Similarly, if we translate the circle centered at **C2**
to be centered at ** P**, then it intersects the red line at a
point ** R** inside the circle centered at ** C1**.
Any point on the perpendicular bisector of **RC2** (shown in red)
is the same distance from **C2** as it is from ** R**.
At the intersection **Q** of the two red lines, since the circle
centered at **C2** and the circle centered at ** P**
have the same radius, **QS** is congruent to **QP**,
and since **C2**, **S**, and ** Q** all
are on the same line, the circle centered at **Q** and the circle
centered at **C2** are both tangent to this circle.

**The locus of Q**

Notice that in both cases, when **Q** is between
**R** and **C1**, **length(RC1) = length(RQ)
+ length(QC1) = length(QC2) + length(QC1)**. As ** P** moves
around the circle centered at ** C1**, **length(RC1)**
remains constant, which means that **length(QC2) + length(QC1)**
also remains constant. By the definition of ellipse, the locus of **Q**
as **P** moves around the circle centered at **C1**
must be an ellipse.

If **Q** is not between **R** and **C1**,
then **length(RC1) = length(QC1) - length(QC2)** or **length(RC1)
= length(QC2) - length(QC1)**, in which case the locus of **Q**
is a hyperbola. This GSP file constructs the locus of
**Q** for circle 1 and circle 2.

What happens if **Q = C1** or **Q = C2**?