Circle bitangent angles Say we have two circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$, respectively. Let their centers be $d$ units apart. There are 4 bitangents, two outer and two inner.
Examine the intersection of an inner bitangent with an outer one, near $C_1$. I'm looking for the value of the angle in the quadrant that contains $C_2$ in terms of $r_1$, $r_2$, and $d$.
Anyone have a reference (or a solution) for this? I'm sure it's been looked at before, but I don't know where to look.
 A: 
I'm going to assume $r_2>r_1$, as in my diagram.  I haven't checked to see whether this is necessary for the rest of my work or not.  This might get confusing, so let me start with a rough outline.  $PX_2C_2I_2$ is a quadrilateral with right angles at $X_2$ and $I_2$ and the desired angle is the interior angle of this quadrilateral at $P$.  I'm going to find the measure of the internal angle at $C_2$ by finding the measures of $\angle I_2C_2C_1$ and $\angle X_2C_2C_1$.
Let's start with $\angle X_2C_2C_1$.  Consider $\triangle C_1C_2R_x$.  It has a right angle at $R_x$, $C_1C_2=d$, and $C_2R_x=r_2-r_1$, so $$\cos(\angle X_2C_2C_1)=\frac{C_2R_x}{C_1C_2}=\frac{r_2-r_1}{d}$$ and $$\angle X_2C_2C_1=\arccos\left(\frac{r_2-r_1}{d}\right).$$
Now, turn to $\triangle C_1C_2R_i$.  It has right angle at $R_i$, $C_1C_2=d$, and $C_2R_i=r_1+r_2$, so $$\cos(\angle I_2C_2C_1)=\frac{C_2R_i}{C_1C_2}=\frac{r_1+r_2}{d}$$ and $$\angle I_2C_2C_1=\arccos\left(\frac{r_1+r_2}{d}\right).$$
Thus, $$\angle X_2C_2I_2=\arccos\left(\frac{r_2-r_1}{d}\right)+\arccos\left(\frac{r_1+r_2}{d}\right)$$
and $$\angle X_2PI_2=180°-\arccos\left(\frac{r_2-r_1}{d}\right)-\arccos\left(\frac{r_1+r_2}{d}\right).$$
A: Its useful to make use of the duality of projective
geometry which interchanges points and lines in the
plane.  The dual of a smooth curve in projective geometry 
is the collection of tangents to the curve.  The projective 
dual to a conic section is a conic section.  Conic sections 
have degree 2 so two conic sections intersect in 2*2=4 points.  Thus there are always 4 bitangents to a pair of conic sections.
We have to take into account complex intersections, 
intersections at infinity, and the multiplicity of a
intersections to account for all 4 bitangents.  For instance
the x-axis counts as 2 bitangents to the parabolas y=x^2 and
y=2*x^2.  The line at infinity counts as the other 2 bitangents.
