An equilateral triangle \(ABC\) has side length \(6\) units. The three altitudes of the triangle meet at \(N\). Show that \(AN=2\sqrt{3}\) units.

This triangle is the base of a pyramid whose apex \(V\) lies on the line through \(N\) perpendicular to the plane \(ABC\). Given that \(VN=2\) units, prove that \(\widehat{VAN}=30^\circ.\)

The perpendicular from \(A\) to the edge \(VC\) meets \(CV\) produced at \(R\). Prove that \(AR=\frac{3}{2}\sqrt{7}\) units, and find the exact value of \(\cos \widehat{ARB}\).