Is there any solution for converting any irregular shape to any constructible regular polygon without changing area through pure construction?

Question

Is there any solution for converting any irregular shape to any constructible regular polygon without changing area through pure construction?

Answer 1

Starting from a polygon having $n$ sides, we can construct an equivalen polygon having $n-1$ sides.

Draw diagonal $DF$ and parallel line to $DF$ passing through $E$. This parallel intersects $CD$ in $H$. The triangles $FDE$ and $FDH$ are equivalent because they have the same base and the same height. So the polygon $ABCHFG$ is equivalent to $ABCDEFG$ and has one side less.

This construction can ideally be done until we get a triangle equivalent to the starting polygon.

A rectangle equivalent to the triangle can be easily constructed.

See the second picture below. $D$ is the midpoint of $AB$, $FE\parallel AB$ and lines $AF,DE$ are perpendicular to $AB$.

A square equivalent to the rectangle $ABCD$ can be constructed in the following way (see third picture below).

With center in $B$ draw an arc having radius $BC$ which intersects the line $AB$ in $F$.

Let $M$ be the midpoint of $AF$. With center in $M$ and radius $AM$ draw a semicircle that intersects the line $CB$ in $H$. The $AHF$ is a right triangle and $BF\cdot AB=BH^2$.

So the red square is equivalent to the blue rectangle.