The angle between two oriented circles at a point of intersection is defined as the angle between the tangents at that point, equipped with the same orientation. Prove by analytic reasoning, rather than geometric inspection, that the angles at the two points of intersection are opposite to each other.
This problem is from Ahlfors' Complex Analysis, page 83, section 3.3.4.3. This section mainly discusses oriented circles, preceded by discussions on linear fractional transformations and cross-ratios. https://mccuan.math.gatech.edu/courses/6321/lars-ahlfors-complex-analysis-third-edition-mcgraw-hill-science_engineering_math-1979.pdf
my attempt:
Let the two circles have their centers at points $A$ and $B$ with radii $r_1$ and $r_2$ respectively. Let $z_1$ and $z_2$ be the complex coordinates of points $A$ and $B$, and let $z$ be the complex coordinate of a point on the first circle. Then, the equation of the two circles can be written as:
\begin{align} (1) \quad |z - z_1| &= r_1 \\ (2) \quad |z - z_2| &= r_2 \end{align}
Now, let $P$ and $Q$ be the two intersection points of these circles. We can represent the tangents at these points using their complex derivatives.
To find the complex derivatives, we first need to write equations (1) and (2) in terms of complex functions. We can achieve this by squaring both sides of each equation:
\begin{align} (1') \quad (z - z_1)(\overline{z} - \overline{z_1}) &= r_1^2 \\ (2') \quad (z - z_2)(\overline{z} - \overline{z_2}) &= r_2^2 \end{align}
where $\overline{z}$ denotes the complex conjugate of $z$.
I don't know what to do next.
