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The Bolyai-Gerwien theorem states:

Given two polygons with the same area, it is possible to cut up one polygon into a finite number of smaller polygonal pieces and from those pieces assemble into the other polygon.

Then, most proofs I've seen go something like this:

One possible proof is to show that any polygon can be broken down into triangles. These triangles can be reassembled into rectangles, which can then be joined into one big rectangle. This big rectangle can then be reassembled into a square.

My question is: how does this prove the theorem? It is indeed true that we can turn some polygon $P$ into a square $S$ of the same area, and then cut $S$ into another set of pieces and reassemble them into some other polygon $P'$ of the same area. But how is the condition in the statement of the theorem that I've made bold really hold? Pieces used to reassemble $P$ into $S$ are not the same pieces used to reassemble $S$ into $P'$. What is going on here?

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    $\begingroup$ $S$ decomposes into a set of pieces that make $P$, and it decomposes into a set of pieces that make $P^\prime$. If you superimpose the dissection cuts from the two sets of pieces created from $S$, you'll arrive at a common set of pieces that can assemble into $P$ or $P^\prime$. $\endgroup$ – Blue Jul 6 '14 at 15:35
  • $\begingroup$ What is this procedure, superimposing the dissection cuts? Google search returned nothing, neither do I understand it myself. $\endgroup$ – user132181 Jul 6 '14 at 15:38
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    $\begingroup$ There's a set of cuts that decomposes square $S$ into pieces that form $P$; there's a second set of cuts that decomposes square $S$ into pieces that form $P^\prime$. Make both sets of cuts to square $S$. The original pieces that formed $P$ divide into sub-pieces, but those sub-pieces together still form $P$ (because the sub-pieces form the pieces that form $P$!); likewise, the pieces that formed $P^\prime$ will be cut into sub-pieces, which still form $P^\prime$. The same set of sub-pieces, then, form $P$ or $P^\prime$. $\endgroup$ – Blue Jul 6 '14 at 15:50
  • $\begingroup$ @Blue thanks for a thorough explanation, it helped a lot! Now I get it completely. Please, post your first and second comments together as an answer so I can accept it :) $\endgroup$ – user132181 Jul 6 '14 at 15:52
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Square $S$ decomposes into a set of pieces that make polygon $P$, and it decomposes into a set of pieces that make $P^\prime$. If you superimpose the dissection cuts from the two sets of pieces created from $S$, you'll arrive at a common set of pieces that can assemble into $P$ or $P^\prime$.

In other words (although pictures would be better!) ...

There's a set of cuts that decomposes square $S$ into pieces that form $P$; there's a second set of cuts that decomposes square $S$ into pieces that form $P^\prime$. Make both sets of cuts to $S$. The combined cuts divide the original pieces that formed $P$ divide into sub-pieces, but those sub-pieces together still form $P$ (because the sub-pieces form the pieces that form $P$ !); likewise, the combined cuts divide the pieces that formed $P^\prime$ into sub-pieces that still form $P^\prime$. The same set of sub-pieces, then, form both $P$ and $P^\prime$.

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