Regional problem · Geometry

A point \(D\) is chosen on the side \(BC\) of a triangle \(ABC\). Points \(E\) and \(F\), both different from \(D\), are chosen on the line \(BC\) so that

BE = BD \qquad \text{and} \qquad CF = CD .

The circles circumscribed about the triangles \(ACE\) and \(ABF\) meet for the second time at a point \(G\). Let \(H\) be the centre of the circle circumscribed about the triangle \(EGF\). Prove that the midpoint of the segment \(DH\) is the centre of the circle circumscribed about the triangle \(ABC\).

\(E\) and \(F\) lie on the line \(BC\) with \(BE = BD\) and \(CF = CD\); the circles through \(A, C, E\) and through \(A, B, F\) meet again at \(G\).