Given points $A$ and $B$, what curve does the locus of points that form congruent angles take?

Question

In more mathematical terms, let $X$ be the set of points that satisfy the following condition: $\forall \, C \in X, \, m \angle ACB = k$ where $k$ is some positive real value less than $180^{\circ}$. Depending on the angle, this set will look different. It's well-known that for $k=90$, $X$ would be a circle with diameter $\overline{AB}$. But for everything that's not 90, I have no idea what that curve would look like.

Answer 1

$X$ looks like an 8 or like an American football (depending on $k$). Just find one such point $C$ and draw the circumscribed circle of $ABC$. Then the arc of that circle from $A$ to $B$ that contains $C$ is part of your $X$. This is so because $\angle ACB=\frac 12\angle AOB$ (where $=$ is the center of the circle). The rest of the curve is obtained by reflecting that arc at $AB$. (And whether or not $A,B$ themselves belong to $X$ is at least doubtful).