In the triangle \(ABC\) the point \( J\) is the center of the excircle opposite to \( A\). This excircle is tangent to the side \(BC\) at \( M\), and to the lines \( AB\) and \(AC\) at \(K\) and \(L\) respectively. The lines \(LM \) and \(BJ\) meet at \(F\), and the lines \(KM \) and \(CJ \) meet at \(G\). Let \(S\) be the point of intersection of the lines \(AF\) and \( BC\), and let \(T\) be the point of intersection of the lines \(AG\) and \(BC\). Prove that \(M\) is the midpoint of \(ST\).