Consider, for the sake of simplicity, a circle $C$ centered at he origin with radius $a$. Let $F=(h,k)$ be a point not necessarily inside the circle. Let $M=(a\cos\theta,a\sin\theta)$ be a point in the circle, and $L$ the line that pases through $M$ and $F$. Let $L_1$ be the line passing through the midpoint of $MO$ where $O$ is the origin, perpendicular to $MO$. Then the curve sought is the locus of points of the point of intersection of $L$ and $L_1$.
You can check that these two lines can be written as
$y = - \frac{1}{{\tan (t)}}\left( {x - \frac{{a\cos \left( t \right)}}{2}} \right) + \frac{{a{\text{ }}\sin (t)}}{2}{\text{ }}$
and
$y = \frac{{a\sin (t) - k}}{{a\cos (t) - h}}{\text{ }}(x - h){\text{ }} + k$
This, what we want to is express $x$ and $y$ in terms of $t$ and the curve will be $(x(t),y(t))$. Now, although the plotting soft produces the right curves, I have failed $4$ times already to solve for $x$ and $y$. I am getting
$x(t) = \frac{{2a\cot t\cos t + 2a\sin t + {\text{2}}f\left( t \right)h - 2k}}{{{\text{ 2}}f\left( t \right) + 2\cot t }}{\text{ }}$
and
$y(t)=f(t)(x(t)-h)+k$
where ${\text{ }}\frac{{a\sin (t) - k}}{{a\cos (t) - h}} = f\left( t \right)$
but this fails to give the correct parametrization.
Could you help to find a closed expression in terms of $t$ and verify it is correct?
These are pics of the curves generated. If the focus is inside the cirlce of radius $a/2$ we get a deformed circle, if it is in the half outside, we get a closed intersecting curve, if it is exaclty on the circle we get the Foluim of Descartes/Trisectrix of Maclaurin, if it is right in the origin, another circle, if it is exactly halfway, a cardioid, and if it is outside the circle, a distorted Foluim of Descartes/Trisectrix of Maclaurin with a part of an hiperbola.