This puzzle is a really tough one. Follow along with the geometric construction below. We are presenting it as a sequence of diagrams each built off of the previous one. In the final diagram we'll derive a fact that just cannot be...

Your job is to figure out what's wrong. Construct line segment AE which is a copy of the left side of the rectangle, rotated by a small angle about of its upper endpoint. Next, construct line segment EC by connecting the lower endpoint of the rotated line segment to the bottom right corner of the rectangle. Construct two perpendicular bisectors. One will bisect AB, the upper edge of the original rectangle. The other will bisect EC, the angled edge created in the previous stage. Label the intersection of these bisectors I. Finally, we construct two triangles. Both involve the point I, the intersection of the perpendicular bisectors created in the previous stage. The red triangle has vertices IBC. The green triangle's vertices are IEA. Note that the two triangles are similar (via a reflection) because of the side-side-side rule. The sides (AI and IB) with a single hash mark in the diagram are equal since the triangle AIB is isosceles. Similarly, the sides (IE and IC) with two hash marks are equal because triangle IEC is also isosceles. The sides with three hash marks (BC and AE) are equal since they are both one of the dimensions of the original rectangle.

Now consider the angles IBC and IAE. Since these are corresponding angles in similar triangles they must of course be equal, yet it is apparent that they differ by the small angle that we rotated the line segment AE by in the beginning of this construction ????