Let see the solution presented at
http://www.math.utk.edu/~freire/teaching/m231f08/m231f08brachistochrone.pdf
Page 4, he has 3 integrations constants : $C, x_{\phi_0}$ and $y_{\phi_0}$ to integrate $x(\phi),y(\phi)$.
He set $x_{\phi_0}=0$ and $y_{\phi_0}=C\,/\,2$ but I don't understand its explanatation that is :
$x=0$ corresponds to $y=0$ hence $\phi=0$.
It seems he imposes that $\phi(0)=0$ which should be proved ?