The "classic textbook proof" of equality of base angles of an isosceles triangles which I studied in my school days is as follows:

Let $\Delta ABC$ be a triangle with $AB = AC$ and let $D$ be the mid point of $BC$. Now the triangles $ADB$ and $ADC$ are congruent via $SSS$ criterion. Hence $\angle ABD = \angle ACD$ and this is the same as $\angle ABC = \angle ACB$.

I studied another proof some years ago in the Douglas R. Hofstadter's famous masterpiece Godel, Escher, Bach which was discovered by a computer program:

Consider the triangles $\Delta ABC, \Delta ACB$. These are congruent via $SSS$ criterion and hence $\angle ABC = \angle ACB$.

This one line proof is amazing, but at the same time looks confusing (some might think it is fishy but it is not) because the original triangle has been mirrored to exchange vertices $B, C$ and then considered congruent to original one.

My question is a pedagogic one:

Is this proof simpler to understand (compared to the standard version) for students of age 12-13 years? Should textbooks at least include this proof as well in some supplement or appendix?

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    $\begingroup$ Not, in my personal opinion, the proof is crisp enough and not as convoluted as you have featured it to be sir. $\endgroup$ Aug 7, 2015 at 10:12
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    $\begingroup$ Definitely interesting in its own right, but the I think the canonical one is easier to understand. $\endgroup$
    – ignoramus
    Aug 7, 2015 at 10:20
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    $\begingroup$ Note that $\triangle ABC$ and $\triangle ACB$ are more-immediately congruent via the SAS postulate. As Wikipedia notes, this version of the one-line proof dates back to Pappus of Alexandria. To your question: Every student should see that proof. Of course, other standard approaches are valuable when starting-out in Geometry, as they are straightforward applications of techniques that students are just learning to use. The one-liner's cleverness is something of an aberration. A good textbook could/should find room for everything. $\endgroup$
    – Blue
    Aug 7, 2015 at 15:28
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    $\begingroup$ @Blue: thanks for the feedback. $\endgroup$
    – Paramanand Singh
    Aug 8, 2015 at 1:58
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    $\begingroup$ I think this proof (and the SAS of @Blue) are confusing, because (of course) they require reusing the same thing differently. At least, I find it confusing, so I hope 12-13yos aren't too far ahead of me. But it is also an opportunity to teach the general technique of seeing the same thing in different ways, and tools to assist (e.g. two copies of the figure?) - very powerful, and probably needs a fair bit of time. The $law of sines$ is similar. If this skill/technique could also be taught, then it would be great to include it. It could also be an optional, standalone, one-off trick. $\endgroup$ May 13, 2018 at 4:57

1 Answer 1


That is a good idea. pay attention for postulates that you are accepted for doing proof. If students are familiar with postulates then it is very suitable to understand.


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