Why the medians of a triangle meet

Points \(A\), \(B\) and \(C\) have position vectors \(\mathbf a\), \(\mathbf b\) and \(\mathbf c\) relative to an origin \(O\).

1. The point \(P\) lies on \(AB\) with \(AP:PB=\lambda:\mu\), where \(\lambda,\mu>0\). Show that \(\overrightarrow{OP}=\dfrac{\mu\mathbf a+\lambda\mathbf b}{\lambda+\mu}\).[3]
2. \(M\) is the midpoint of \(BC\), and \(G\) lies on \(AM\) with \(AG:GM=2:1\). Show that \(\overrightarrow{OG}=\dfrac13(\mathbf a+\mathbf b+\mathbf c)\).[2]
3. Explain why the three medians of the triangle all pass through \(G\).[2]
4. Find the coordinates of \(G\) when \(A=(1,2)\), \(B=(7,4)\) and \(C=(4,9)\).[1]