Regional problem · Geometry

Let \(T\) be the centroid of an acute triangle \(ABC\). Let \(A'\) be the foot of the altitude drawn from \(A\) to the side \(BC\), and let \(A''\) be the point of the segment \(BC\) for which

\[ BA' = A''C . \]

The ray \(AA''\) meets the circumcircle of the triangle \(ABC\) once more, at a point \(M\), and the ray \(TA'\) meets that circle at a point \(N\). Prove that \(MN \parallel BC\).

The configuration: \(A'\) and \(A''\) on \(BC\), the centroid \(T\), and the two rays cutting the circumcircle at \(M\) and \(N\).