Let's construct a third polygon similar to two given polygons, whose area is the sum of the areas of the previous two.
Let the areas of the two given similar polygons be $t_{1}$ and $t_{2}$, and one of their corresponding sides be $a_{1}$ and $a_{2}$.

It is known that

$$
t_{1}: t_{2}=a_{1}^{2}: a_{1}^{2}
$$

which implies that

$$
t_{1}: t_{2}:\left(t_{1}+t_{2}\right)=a_{1}^{2}: a_{2}^{2}:\left(a_{1}^{2}+a_{2}^{2}\right)
$$

This means that the side of the sought third polygon is the hypotenuse of a right-angled triangle whose legs are $a_{1}$ and $a_{2}$. Constructing the hypotenuse and then constructing a polygon similar to the given polygons over this hypotenuse is already trivial.

Béla Kovács (Sárospatak, Rákóczi g. II. o. t.)