IMO 1963 Problem 3

In an $n$ -gon $A_{1}A_{2}\ldots A_{n}$ , all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation $$a_{1}\geq a_{2}\geq \dots \geq a_{n}.$$ Prove that $a_{1}=a_{2}= \ldots= a_{n}$ .