City problem · Geometry

Let \(ABC\) be a triangle. On side \(AB\) choose points \(C_1\) and \(C_2\) with

AC_1 = \tfrac{2015}{3015}\,AB, \qquad AC_2 = \tfrac{2015}{3014}\,AB;

on side \(BC\) choose points \(A_1\) and \(A_2\) with

BA_1 = \tfrac{1007}{2015}\,BC, \qquad BA_2 = \tfrac{1008}{2015}\,BC;

and on side \(AC\) choose points \(B_1\) and \(B_2\) with

AB_1 = \tfrac{2015}{3031}\,AC, \qquad AB_2 = \tfrac{2015}{3030}\,AC.

The line \(AA_1\) meets the lines \(B_1C_1\) and \(B_2C_2\) at \(M\) and \(N\) respectively, and the line \(AA_2\) meets \(B_1C_1\) and \(B_2C_2\) at \(Q\) and \(P\) respectively. Prove that the centroid of triangle \(ABC\) lies inside the quadrilateral \(MNPQ\).