This proof can't possibly be right.

So...what's wrong with it?

We start with an ordinary rectangle.                              



We swing one edge out by some angle, without changing its length. It doesn't matter so much what angle we use, but this looks like about 30 degrees:



We connect the free end of that edge back to the indicated vertex of the rectangle:



We draw (1) the perpendicular bisector of the most recently added edge, and (2) the perpendicular bisector of the top edge of the rectangle. They must cross somewhere, because they are not parallel. I've drawn the case where they cross above the top edge of the rectangle; you might also draw the case where they cross on the top edge and the case where they cross below the top edge. Don't worry: we reach the same disaster in those cases as in the case I've drawn.



Ignoring the bottom-left corner of the rectangle, connect all other vertices to the point of intersection.



Since this special point lies on the perpendicular bisector of the top edge of the rectangle, this shaded triangle is isosceles:



And likewise, since the point lies on the perpendicular bisector of that sloping edge, this shaded triangle is also isosceles.



So, the first shaded triangle tells us that the blue edges in this diagram are the same length, while the second triangle tells us that the red edges are the same length. Also, because the edge we swung out at the very beginning is the same length as the side of the rectangle, the green edges are the same length. By Side-Side-Side, the two red/blue/green triangles are congruent.



Since those triangles are congruent, we may conclude that the angles between Blue and Green edges are congruent. Furthermore, the two angles that the blue edges make with the vertical edges of the rectangle are also congruent. Thus, all the angles marked in black are congruent. Since the two angles in the upper left corner are congruent, their difference (marked in red) must measure zero degrees.



That means that no matter how much of an angle we used at the beginning, the angle turns out to be zero degrees. Or, put differently, every number is zero.
Right?


I hope you are not convinced that every number is zero; instead, you should realize that the above proof is HORRIBLY WRONG.
I challenge you to find the error in the proof.

This construction was first proposed by Monsieur Marty, Ecole Nationale Professionelle de Tarbes, France.