Show that the marked triangle is isosceles



The symmetry at the perpendicular bisector of the segment \(AB\) 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.
Since the symmetry swaps the segments \(AC\) and \(BC\), these have the same length.