Show that the marked triangle is isosceles



The triangle is isosceles because it has a symmetry axis.
The symmetry at the perpendicular bisector of the segment \(AB\) clearly swaps \(A\) and \(B\).
Moreover, the point \(C\) stays fixed (because it is the intersection of two arcs that are mapped one into the other by the symmetry).
So the symmetry leaves the triangle \(ABC\) invariant.