Reference for a dual isoperimetric problem and solution I am trying to track down the first published solution to the following problem:
What curves within the unit disc in the plane and endpoints on the unit circle, minimize their length (within the ball) while dividing the area of the ball into two regions of a given ratio?
The solution is, of course, circular arcs orthogonal to the unit circle.

The only complete solution and proof that I've been able to find is embedded in a much stronger theorem within "Stability for Hypersurfaces of Constant Mean Curvature with Free Boundary" by A. Ros and E. Vergasta, Geometriae Dedicata, volume 56, 1995.  But I suspect that there is an earlier proof for this specific case.
If anyone can point me in the right direction, I would very much appreciate it.
Edited for clarity of problem.
 A: I think that a straight variational approach   has a good chance of yielding the desired conclusion.   Fix two points $z_0,z_1$  on the boundary of the unit disk. Denote by $C$ the positively oriented  arc of the circle that runs from $z_0$ to $z_1$.
Let  $\mathcal{P}_{z_0,z_1}$ the set of paths $[0,1]\to \mathbb{C}$        with endpoints $z_0$ $z_1$.   $\mathcal{P}_{z_0,z_1}$ is an affine space modeled on the vector space of  maps  $[0,1]\to\mathbb{C}$ that vanish at  endpoints.
If $\gamma\in \mathcal{P}_{z_0,z_1}$ is  an embedded  path inside the  disk, then the area  between    $\gamma$ and  $C$  is given by the integral
$$\int_C xdy -\int_\gamma xdy $$
Your constraint  can now be given a simpler form
$$\int_\gamma xdy =const. $$
Now   solve the   constrained   variational problem
$$ \min\left\lbrace \int_0^1   |\dot{\gamma}(t)| dt;\;\;\gamma\in \mathcal{P}_{z_0,z_1},\;\;\int_\gamma xdy =const\right\rbrace. $$
If we write $\gamma(t)= x(t) + i y(t)$ then the constraint equation      can be rewritten as
$$ \int_0^1 x(t) \dot{y}(t) dt =const. $$
This    defines a  quadratic hypersurface in the affine space $\mathcal{P}_{z_0,z_1}$.
If we define
$$L, F: \mathcal{P}_{z_0,z_1}\to \mathbb{R}, $$
$$ L(\gamma)=  \int_0^1   |\dot{\gamma}(t)| dt,\;\; F(\gamma)= \int_\gamma xdy$$
then the Euler-Lagrange equations for the above variational problem have the form
$$ dL  +\lambda dG=0 \tag{$EL_\lambda$}$$
where $\lambda\in\mathbb{R}$ is  a Lagrange multiplier  and $dL$ and $dF$ are the differentials  of $L$ resp. $F$.  The differential of $F$ is   a linear function so ($EL_\lambda$) can be viewed as a nonlinear eigenvalue problem.        It can be written very explicitly and I believe   that playing with it  will yield   the desired conclusion.
