Great Circle on Sphere $S^2$ Question / Task / Problem
$a)$ Calculate all points $E\in S^2$ so that $d_s(A,E)= d_s  (B,E)=\pi/3$
$b)$ Calculate the cosines $\cos (a)$, $\cos (b)$, $\cos (c)$ of the side lengths in the triangle $(A,B,C)$. Which of the side lengths is the largest, which is the smallest?
The points $A$, $B$, $C$ are defined as follows:
$A=\frac{1}{3}\cdot (1,2,2)$; $B=\frac{1}{3}\cdot(1,-2,2)$; $C=\frac{1}{\sqrt{2}}\cdot(0,1,1)$
In the subtask $a)$ I have absolutely no idea. 
In subtask $b)$ I know that the side lengths are so defined:
\begin{align*}
\cos (a) & = \cos (b) * \cos (c) + \sin (b) * \sin (c) * \sin (\alpha)
\\ \cos⁡(b) & = \cos⁡(a)  * \cos (c) + \sin (a) * \sin (c) * \sin (\beta)
\\ \cos (c) & = \cos (a) * \cos (b) + \sin (b) * \sin (a) * \sin (\gamma)
\end{align*}
But that does not help me. 
I know that the points $A$, $B$ and $C$ are the intersections of Great circles. Spontaneously, I would think of the cutting angle. But how is this calculated?
Or I have read that the calculation about the center of the sphere is possible. But I do not know that either. I am frustrated. 
What do I overlook?
Could someone please help me!
 A: (a) Take cosines of $d_s(A,E)=d_s(B,E)=\pi/3$ to get $A\cdot E=B\cdot E=\cos(\pi/3)$. Write $E=(x,y,z)$ so this becomes a linear system of equations defining a line:
$$ \begin{cases} \frac{1}{3}x+\frac{2}{3}y+\frac{2}{3}z=\frac{1}{2} \\ \frac{1}{3}x-\frac{2}{3}y+\frac{2}{3}z=\frac{1}{2} \end{cases} $$
Eliminating yields $y=0$ and $x=\frac{3}{2}-2z$. Thus the line is parametrized by $(\frac{3}{2}-2t,0,t)$. There will be two values of $t$ for which such a point is on the sphere.
(a') Alternatively, one can note that any point equidistance from $A$ and $B$ will be on the great circle exactly between them, which is the sphere intersected with the plane spanned by the midpoint $u=(A+B)/2$ and the cross product $v=A\times B$. So you can normalize and then get solve for $\phi$ for which
$$ \big[(\cos\phi) \hat{u}+(\sin\phi)\hat{v}\big]\cdot A=\cos(\pi/3). $$
Note $\hat{v}$ is perpendicular $A$ and $B$, and you can explicitly calculate $\hat{u}\cdot A$.
(b) $\cos(a)$ is the dot product $B\cdot C$, etc.
