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 the perpendicular bisector of the most recently added edge, and 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 should also draw the case where they cross on the top edge and also the case where they cross below the top edge. Don't worry:
we reach the same disaster in those cases.
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 below 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. Also, 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. In particular, 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?
This construction was first proposed by Monsieur Marty, Ecole Nationale Professionelle de Tarbes, France.
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