attributed to Lewis Carroll
Consider a triangle ABC. Bisect BC, and call the point of bisection D. Construct a perpendicular from this point. Bisect angle BAC. Find the point where the angle bisector intersects the perpendicular constructed previously, and call it E.
Now construct a line from E perpendicular to AB. The point where it meets AB will be called F. Do the same on the other side: construct a line from E perpendicular to AC, and call the point of intersection G.
Now, ED is congruent to itself, and the angle EDB and EDC were both constructed as right angles, and DB is congruent to DC because D was chosen to bisect BC. Thus, the triangles EDB and EDC are congruent.
The angles EFA and EGA were both constructed as right angles, the angles FAE and GAE are the halves of a bisected angle, and of course AE is congruent to itself. Thus, the triangles FAE and GAE are congruent.
EB is congruent to EC, because they are corresponding parts of the congruent triangles EDB and EDC. Likewise, FE and GE are corresponding parts of FAE and GAE. Since the angles BFE and CGE are right angles, it follows that FEB and GEC are congruent (angle-side-side sufficing in the special case of a right angle).
Now, AF and AG are corresponding parts of FAE and GAE, and FB and GC are corresponding parts of FEB and GEC. Since AF=AG and FB=GC, AF+FB=AG+GC. Thus, AB=AC.
As a corolloary, it's simple to show that all triangles are equilateral, because it can be shown by the same method that AB=BC and BC=AC.
So, what's wrong with the proof? You won't find anything wrong
by going over the above description step-by-step. Every step is valid. Even the angle-side-side part, which tends to alarm students who remember it as the one triplet that doesn't work for proving congruence of triangles, is in fact valid when the angle is a right angle. If you want to work out the error
for yourself, stop reading now.
The problem is in the unstated assumption that F is between A and B, and G is between A and C. In a scalene triangle, these will not both be true. One of them will lie outside the triangle altogether. If it's F, then AB will be the difference between AF and FB, not the sum. E will also lie outside the triangle, but this does not affect the proof.