Area Problems by Rearranging
A number of DPWW are based on the additivity property of areas. In these proofs, a region is cut up into parts, which are then rearranged to form a better known shape. Even though these proofs can be very elegant, without proper justification of the constructions they can be misleading. A famous example of this is the "Missing Square" paradox.
Example 3. The Area of the Regular Dodecagon
This example is based on the well-known Kürschák’s tile. The dynamic rearrangement of the tiles makes a convincing case that the area of the dodecagon is ¾ of the area of the circumscribed square, but this proof should be supplemented by a few lines of calculations showing that certain triangles are congruent.
More details.
Example 4. The midsegments divide any triangle in four congruent triangles.
Other DPWW are completely self-explanatory and do not need any additional calculations.
One such construction can be seen in the example below. Here the proof is based on properties of the medians and the fact that rotations preserve the area.
More details.