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 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

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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.

  • $\begingroup$ Why not use the line of the intersecting points as a reference and use this line to cut the plane of the two circles (assuming they are in the same plane) into right and left and show that one angle points left and the other right - the directed circles will support the solution. $\endgroup$
    – Moti
    Apr 24 at 18:31
  • $\begingroup$ @Moti So, how should I algebraically represent this line? After that, do the following steps involve checking the signs of the imaginary parts of the points on the left and right sides of this line or something else? Do I need to algebraically express this angle? $\endgroup$
    – liyushu
    Apr 25 at 11:29


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