I have two questions.
First, is there a relatively simple formula for the midpoint of two points $a_1$ and $a_2$ in the disk with respect to the hyperbolic geometry? That is, the point on the bisector of $a_1$ and $a_2$ which has minimal distance from $a_1$ and $a_2$.
Second, if $B$ denotes the degree $2$ finite Blaschke product with zeros $a_1$ and $a_2$, is the critical point of $B$ which is in the disk equal to the midpoint of $a_1$ and $a_2$?
Thanks
Critical point
Yes, it's at the midpoint. For Blaschke products $B$ with zeros at $\pm a$ this is immediate from the formula $B(z)=(z^2-a^2)/(1-\bar a^2 z^2)$.
If zeros are elsewhere, say $z_1,z_2$, then consider the composition $B\circ \phi^{-1}$ where $\phi$ is a Möbius transformation that sends the hyperbolic midpoint between $z_1,z_2$ to $0$. This composition is a Blaschke product with zeros $\phi(z_1),\phi(z_2)$ which satisfy $\phi(z_1)+\phi(z_2)=0$, reducing the problem to the previous case.
Midpoint
The connection with Blaschke product might be the easiest approach. Move one of points to zero, the other goes to
$$ c = \frac{a_1-a_2}{1-a_1\bar a_2}$$ The midpoint between $0$ and $c$ can be found by differentiating $z(z-c)/(1-\bar cz)$. When $c$ is between $0$ and $1$ (can be achieved by rotation), it's $$ d = \frac{1-\sqrt{1-c^2}}{c} $$ Finally, the midpoint is obtained by undoing, $$ m = \frac{d+a_2}{1+m\bar a_2} $$