If the ratio of areas of two polygons is the square of the ratio of their perimeters, are the polygons similar? It is known that for two similar polygons $A$ and $B$, the ratio of the areas is the square of the ratio of the perimeters. That is, for example, if the ratio of perimeter A to perimeter B is 5:7, then the ratio of their areas is 25:49. However, is the converse to this statement true? That is,

If the ratio of areas of two polygons is the square of the ratio of their perimeters, are the polygons similar? What if we add the specification that the polygons have the same number of vertices? 

I cannot think of a counterexample. Any help is appreciated!

HINT.-Let $p_1$ be the perimeter and $A_1$ the area of the polygone $P_1$ and similarly $p_2$ and $A_2$ for the polygone $P_2$ with $$\frac{p_1^2}{p_2^2}=\frac{A_1}{A_2}$$ You have $$p_2=s_1+s_2+\cdots+s_n$$ How many changes of sides $s_i$ preserving the value $A_2$ are there? The answer to your problem is NO.
