Imagine any two fixed points $A$, $B$ inside a circle, and add a third point $C$ on the line segment $\overline{AB}$.
We may write
$$ \vec{c}=\vec{a}+\mu(\vec{b}-\vec{a}) $$
and name $a$, $b$ and $c=c(\mu)$ the distance of each of those points from the center of the circle.
I'm looking for any simple relation of $c$ in terms of $\mu$ and given $a$ and $b$, because what I could come up with so far does look far more complicated than one might expect, so maybe I'm missing something.
From law of sines:
$$ c = \mu(1-\mu)d\left(\frac{\sin\alpha}{\sin\delta_1}+\frac{\sin\beta}{\sin\delta_2}\right) $$
(actually looks quite nice, but $\delta_1$ and $\delta_2$ are dependent of $\mu$).
From law of cosines:
$$ c^2=\tfrac12(d^2\mu^2+d^2(1-\mu)^2+a^2+b^2)-d(a\cos\beta\cdot\mu+b\cos\alpha\cdot(1-\mu)) $$
Note $$ \vec{c}=\vec{a}+\mu(\vec{b}-\vec{a})=(1-\mu)\vec a +\mu \vec b $$
Then
$$c^2 = (1-\mu)^2a^2 +\mu^2 b^2 +2\mu(1-\mu)\vec a \cdot \vec b$$