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    The locus $\Arg\qty{z-a\over z-b}=\theta$: circular arcs through $a$ and $b$

    E-WH2YQ

    Area
    Complex Analysis
    Topics
    Fractional Linear Transformations, Geometry
    Status
    draft

    Fix \(a,b\in \CC\) and \(\theta\), and describe the locus \[ \ts{z\st \Arg\qty{z-a\over z-b} = \theta} .\]

    Solution

    The geometry at hand:

    By the inscribed angle theorem, this locus is an arc of a circle whose center \(O\) is the point for which the angle \(aOb\) is \(2\theta\):

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