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Let points $A,B$ be fixed, and let $C$ vary, with the constraint that $\gamma := \angle ACB$ is fixed. It is well known that the locus of $C$ is a double arc, where $AB$ is a chord of measure $2 * \angle ACB$.

What constraints describe a circle where $AB$ is a chord of measure $\gamma$?

I conjecture that if instead of requiring that $\gamma := \angle ACB$ we require that the oriented angle $\measuredangle ABC$ is fixed to equal $\gamma$ this will describe such a circle. However:

  1. How do we prove this?
  2. What about the points $A$ and $B$ themselves? They seem to be limiting cases that are approached for all $\gamma$ as $C$ approaches $A$ or $B$
  3. Experimenting with this seems to show that in some extreme cases, we get not a circle but a line (which can be thought of as a generalized circle) or a point (a degenerate circle), but I'm having trouble characterizing this
SRobertJames
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  • If $A$ and $B$ are fixed then $AB$ is a chord of a circle whose centre is any point on the perpendicular bisector of $AB$ – Henry Dec 17 '23 at 02:07

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