### Areas

Is there any relationship between the area of a polygon (in the usual sense) and the area of the polygon obtained by the transformations we have been considering? For example, in the case of bisection, we have the following:

- If \(P\) is a triangle, its image is obtained by an homothety of ratio \(-1/2\), thus the area of the obtained triangle is \(1/4\) of the area of \(P\).
- If \(P\) is a quadrilateral, its image is a parallelogram whose area is half of the area of \(P\).

For polygons with a number of vertices greater than 4, the ratio between the areas is not constant; the area of the obtained polygon may even be greater than the area of the initial polygon in case the latter is not convex. However, something curious happens: if there is more than one initial polygon giving rise to the same polygon (which happens when the number of vertices is even), they all have the same area!

**Note:** when the ratio between the areas is not constant, it
is easy to see that the area of the new polygon does not depend only on the
area of the initial polygon. For this, we will assume that we have two initial
polygons \(P\) and \(Q\) that originate, by bisection, the polygons \(P'\) and
\(Q'\), respectively, where the ratio between the areas of \(P\) and \(P'\)
is different from the ratio between the areas of \(Q\) and \(Q'\). Applying
an homothety of ratio \(r\) to \(Q\), we get a new polygon \(R\) which, by bisection,
produces a new \(R'\). The values of the areas of \(R\) and \(R'\) are obtained
by multiplying by \(r^2\) the values of the areas of \(Q\) and \(Q'\), respectively.
However, if \(r^2\) is equal to the ratio between the areas of \(P\) and \(Q\),
we have \[Area(R)\;=\;r^2 Area(Q)\;=\;\frac{Area(P)}{Area(Q)} Area(Q)\;=\;Area(P)\]
\[Area(R')\;=\;r^2 Area(Q')\;=\;\frac{Area(P)}{Area(Q)} Area(Q')\;\neq\;Area(P')\]
(if \(\frac{Area(P)}{Area(Q)} Area(Q')=Area(P')\), we would have \(\frac{Area(Q')}{Area(Q)}=\frac{Area(P')}{Area(P)}\)
and the ratio between the areas of \(P\) and \(P'\) would be equal to the ratio
between the areas of \(Q\) and \(Q'\), impossible by assumption). Therefore,
we have two polygons (\(P\) and \(R\)) with the same area that produce two polygons
(\(P'\) and \(R'\)) with different areas. Hence, the area of the new polygon
does not depend on the area of the initial polygon.