In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the * pons asinorum* (Latin: [ˈpõːs asɪˈnoːrũː], English: /ˈpɒnz ˌæsɪˈnɔːrəm/

*Pons asinorum* is also used **metaphorically** for a problem or challenge which acts as a test of critical thinking, referring to the "ass' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645.^{[2]}

A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem.^{[3]}^{[4]} In fact, Marvin Minsky recounts that he had rediscovered the Pappus proof (which he was not aware of) by simulating what a mechanical theorem prover might do.^{[5]}^{[6]}

Euclid's statement of the pons asinorum includes a second conclusion that if the equal sides of the triangle are extended below the base, then the angles between the extensions and the base are also equal. Euclid's proof involves drawing auxiliary lines to these extensions. But, as Euclid's commentator Proclus points out, Euclid never uses the second conclusion and his proof can be simplified somewhat by drawing the auxiliary lines to the sides of the triangle instead, the rest of the proof proceeding in more or less the same way.

There has been much speculation and debate as to why Euclid added the second conclusion to the theorem, given that it makes the proof more complicated. One plausible explanation, given by Proclus, is that the second conclusion can be used in possible objections to the proofs of later propositions where Euclid does not cover every case.^{[7]} The proof relies heavily on what is today called side-angle-side, the previous proposition in the *Elements*.

Proclus' variation of Euclid's proof proceeds as follows:^{[8]}

- Let
*ABC*be an isosceles triangle with*AB*and*AC*being the equal sides. Pick an arbitrary point*D*on side*AB*and construct*E*on*AC*so that*AD*=*AE*. Draw the lines*BE*,*DC*and*DE*. - Consider the triangles
*BAE*and*CAD*;*BA*=*CA*,*AE*=*AD*, and is equal to itself, so by side-angle-side, the triangles are congruent and corresponding sides and angles are equal. - Therefore and , and
*BE*=*CD*. - Since
*AB*=*AC*and*AD*=*AE*,*BD*=*CE*by subtraction of equal parts. - Now consider the triangles
*DBE*and*ECD*;*BD*=*CE*,*BE*=*CD*, and have just been shown, so applying side-angle-side again, the triangles are congruent. - Therefore and .
- Since and , by subtraction of equal parts.
- Consider a third pair of triangles,
*BDC*and*CEB*;*DB*=*EC*,*DC*=*EB*, and , so applying side-angle-side a third time, the triangles are congruent. - In particular, angle
*CBD*=*BCE*, which was to be proved.

Proclus gives a much shorter proof attributed to Pappus of Alexandria. This is not only simpler but it requires no additional construction at all. The method of proof is to apply side-angle-side to the triangle and its mirror image. More modern authors, in imitation of the method of proof given for the previous proposition have described this as picking up the triangle, turning it over and laying it down upon itself.^{[9]}^{[6]}
This method is lampooned by Charles Lutwidge Dodgson in *Euclid and his Modern Rivals*, calling it an "Irish bull" because it apparently requires the triangle to be in two places at once.^{[10]}

The proof is as follows:^{[11]}

- Let
*ABC*be an isosceles triangle with*AB*and*AC*being the equal sides. - Consider the triangles
*ABC*and*ACB*, where*ACB*is considered a second triangle with vertices*A*,*C*and*B*corresponding respectively to*A*,*B*and*C*in the original triangle. - is equal to itself,
*AB*=*AC*and*AC*=*AB*, so by side-angle-side, triangles*ABC*and*ACB*are congruent. - In particular, .
^{[12]}

A standard textbook method is to construct the bisector of the angle at *A*.^{[13]}
This is simpler than Euclid's proof, but Euclid does not present the construction of an angle bisector until proposition 9. So the order of presentation of the Euclid's propositions would have to be changed to avoid the possibility of circular reasoning.

The proof proceeds as follows:^{[14]}

- As before, let the triangle be
*ABC*with*AB*=*AC*. - Construct the angle bisector of and extend it to meet
*BC*at*X*. *AB*=*AC*and*AX*is equal to itself.- Furthermore, , so, applying side-angle-side, triangle
*BAX*and triangle*CAX*are congruent. - It follows that the angles at
*B*and*C*are equal.

Legendre uses a similar construction in *Éléments de géométrie*, but taking *X* to be the midpoint of *BC*.^{[15]} The proof is similar but side-side-side must be used instead of side-angle-side, and side-side-side is not given by Euclid until later in the *Elements*.

In 1876, while a member of the United States Congress, future President James A. Garfield developed a proof using the trapezoid, which was published in the *New England Journal of Education*.^{[16]} Mathematics historian William Dunham wrote that Garfield's trapezoid work was "really a very clever proof."^{[17]} According to the *Journal*, Garfield arrived at the proof "in mathematical amusements and discussions with other members of congress."^{[18]}

The isosceles triangle theorem holds in inner product spaces over the real or complex numbers. In such spaces, it takes a form that says of vectors *x*, *y*, and *z* that if^{[19]}

then

Since

and

where *θ* is the angle between the two vectors, the conclusion of this inner product space form of the theorem is equivalent to the statement about equality of angles.

Uses of the *pons asinorum* as a metaphor for a test of critical thinking include:

- Richard Aungerville's 14th century Philobiblon contains the passage "Quot Euclidis discipulos retrojecit Elefuga quasi scopulos eminens et abruptus, qui nullo scalarum suffragio scandi posset! Durus, inquiunt, est his sermo; quis potest eum audire?", which compares the theorem to a steep cliff that no ladder may help scale and asks how many would-be geometers have been turned away.
^{[20]} - The term
*pons asinorum*, in both its meanings as a bridge and as a test, is used as a metaphor for finding the middle term of a syllogism.^{[20]} - The 18th-century poet Thomas Campbell wrote a humorous poem called "Pons asinorum" where a geometry class assails the theorem as a company of soldiers might charge a fortress; the battle was not without casualties.
^{[24]} - Economist John Stuart Mill called Ricardo's Law of Rent the
*pons asinorum*of economics.^{[25]} *Pons Asinorum*is the name given to a particular configuration^{[26]}of a Rubik's Cube.- Eric Raymond referred to the issue of syntactically-significant whitespace in the Python programming language as its
*pons asinorum.*^{[27]} - The Finnish
*aasinsilta*and Swedish*åsnebrygga*is a literary technique where a tenuous, even contrived connection between two arguments or topics, which is almost but not quite a non sequitur, is used as an awkward transition between them.^{[28]}In serious text, it is considered a stylistic error, since it belongs properly to the stream of consciousness- or causerie-style writing. Typical examples are ending a section by telling what the next section is about, without bothering to explain why the topics are related, expanding a casual mention into a detailed treatment, or finding a contrived connection between the topics (e.g. "We bought some red wine; speaking of red liquids, tomorrow is the World Blood Donor Day"). - In Dutch,
*ezelsbruggetje*('little bridge of asses') is the word for a mnemonic. The same is true for the German*Eselsbrücke*. - In Czech,
*oslí můstek*has two meanings – it can describe either a contrived connection between two topics or a mnemonic.