Similar polygons can be divided into equal numbers of similar triangles corresponding (in proportion) to the wholes, and one polygon has to the (other) polygon a squared ratio with respect to (that) a corresponding side (has) to a corresponding side. * Let $ABCDE$ and $FGHKL$ be similar polygons, and let $AB$ correspond to $FG$. * I say that polygons $ABCDE$ and $FGHKL$ can be divided into equal numbers of similar triangles corresponding (in proportion) to the wholes, and (that) polygon $ABCDE$ has a squared ratio to polygon $FGHKL$ with respect to that $AB$ (has) to $FG$.
The ratio of the areas of two similar rectilinear figures is proportional to the squared ratio of the lengths of corresponding sides.1
Euclid formulates this proposition generally for all similar rectilinear figures but proves it for pentagonal figures only (editor's note). ↩