54. (SPA 4) Let $P$ be a convex planar polygon with equal angles. Let $l_{1} \ldots, l_{n}$ be its sides. Show that a necessary and sufficient condition for $P$ to be regular is that the sum of the ratios $\frac{L_{1}}{l_{i+1}}\left(i=1, \ldots, n ; l_{n+1}=l_{1}\right)$ equals the number of sides.