Noooooo...
The need to pass the chord \(a'b'\) through the center of the circle as \(ab\) is adjusted to rotate \(a'b'\) means that the closed curve's continuity does not guarantee smooth operation. This is a case of two changing variables; center and angle.
The 'center' was hidden behind the word 'bisect'.
In this particular case, it maybe possible to move point \(b\) closer to \(a'\) or \(b'\) to form a smaller circle that makes \(ab\) and \(a'b'\) bisect (both set of chord new intersections between circle and the closed curve).
No proof.
