Proof 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.