The pizza theorem (Q2740471)

If a circular disk (shortly called a ``pizza'') is cut into \(4n\) slices by \(2n\) concurrent cuts (which run right across the circular disk, having a point \(P\) in common) at equal angles to each other, and \(n\) people share it (the pizza) by taking every \(n\)th slice (thus receiving four slices each) then they receive equal shares. To prove this, the \(k\)th person's share is expressed by a definite integral (using polar co-ordinates with pole \(P\)) the integrand of which turns out to be simply \(2R^2\) (\(R\)= radius of the pizza) if the secant-tangent theorem of circular geometry is applied to its four terms.