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.

Could we draw a picture of \(ABC\) with the altitudes and \(N\) marked? Can you spot any similar triangles? Or right-angled triangles?

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 \(\widehat{VAN}=30^\circ.\)

Could we draw another picture? In three dimensions this time?

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 \(\cos \widehat{ARB}\).

The phrase “\(CV\) produced” means, “the line segment \(CV\) extended”.

Could we draw a picture of \(AVC\), marking in \(R\)? Which lengths in this triangle do we already know?