N1985 P1 Q3

The vertices \(A\), \(B\) and \(C\) of a triangle have position vectors \(\mathbf a\), \(\mathbf b\), \(\mathbf c\) respectively. \(E\) is the midpoint of \(AC\), \(F\) is the midpoint of \(AB\), and the lines \(BE\) and \(CF\) intersect at \(G\). Derive the position vector of \(G\) and prove that \(\overrightarrow{GC}\times\overrightarrow{GB}=\frac13(\mathbf c\times\mathbf b+\mathbf b\times\mathbf a+\mathbf a\times\mathbf c)\).

Hence, or otherwise, prove that the areas of the quadrilateral \(AFGE\) and the triangle \(GBC\) are equal.