Mobius transformation producing a curved triangle with 3 intersecting circles 
Let $ABC$ be a curved triangle on a plane, whose side $AB$, $BC$ and
  $CD$ are arcs of circles $S_1$, $S_2$ and $S_3$ passing though a point $D$ (i.e. $S_1∩S_2∩S_3 = D$, $D≠A$, $D≠B$, $D≠C$). Assume that the angles at the vertices $A$ and $B$ are $30$ degrees each. Find the angle at the vertex $C$.


I really do not understand how to do this. There is a hint to simplify the problem with the appropriate Mobius transformation, but I am completely lost. How are we meant to know what specific Mobius transformation leads to a certain result (in this case, three circles that intersect at one point)? Any help is greatly appreciated. 
By this question Möbius Transformation of Triangles, the answer says that "the image of your triangle is a triangle with circular arc sides, which DO meet at the same angles that the sides of the triangle did"
I am confused by what this means, since it implies that the angle at vertex $C$ is $120$ but curved triangles have a sum of angles $>180$. I am sure it is applicable in some way to my situation, though.
 A: We know that Mobius transformations transform generalized circles into generalized circles. Now, our 3 circles meet at a point which is not on the triangle. If we now think of this point as an element of $\mathbb{C}$ (in the obvious way- we take a point $(a,b)$ to $a+bi$), we can always find a mobius transformation which sends this point $D\in\mathbb{C}$ to infinity- say, $z\mapsto \frac{1}{z-D}$. Note that each of the circles we have transform to a generalized circle, but they all share the point $D$ which is mapped to $\infty$. Hence, they cannot be mapped to circles, and they must be mapped to lines. The 3 arcs we have will be mapped to 3 lines, which have pairwise intersection at certain points- i.e, a triangle. As the transformation is conformal, the angles are preserved at the intersection points- that is, in the vertices of the triangle. So now we can finish, as the sum of angles in a triangle is $180$.
As for what you said in the end, this shows that some curved triangles have angle sum $=180$. Some do not. 
