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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 in a field, separating capable and incapable reasoners; it represents a test of ability or understanding. Its first known usage in this 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*.

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^{[16]}

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.

Another medieval term for the pons asinorum was **Elefuga** which, according to Roger Bacon, comes from Greek *elegia* "misery", and Latin *fuga* "flight", that is "flight of the wretches". Though this etymology is dubious, it is echoed in Chaucer's use of the term "flemyng of wreches" for the theorem.^{[17]}

There are two possible explanations for the name *pons asinorum*, the simplest being that the diagram used resembles an actual bridge. But the more popular explanation is that it is the first real test in the *Elements* of the intelligence of the reader and functions as a "bridge" to the harder propositions that follow.^{[18]} Gauss supposedly once espoused a similar belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.^{[19]}

Similarly, the name *Dulcarnon* was given to the 47th proposition of Book I of Euclid, better known as the Pythagorean theorem, after the Arabic *Dhū 'l qarnain* ذُو ٱلْقَرْنَيْن, meaning "the owner of the two horns", because diagrams of the theorem showed two smaller squares like horns at the top of the figure. The term is also used as a metaphor for a dilemma.^{[17]} The theorem was also sometimes called "the Windmill" for similar reasons.^{[20]}

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

- Richard Aungerville's 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.
^{[17]} - 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.^{[17]} - 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.
^{[21]} - Economist John Stuart Mill called Ricardo's Law of Rent the
*pons asinorum*of economics.^{[22]} *Pons Asinorum*is the name given to a particular configuration^{[23]}of a Rubik's Cube.- Eric Raymond referred to the issue of syntactically-significant whitespace in the Python programming language as its
*pons asinorum.*^{[24]} - 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.^{[25]}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.

**^**Smith, David Eugene (1925).*History Of Mathematics*. Vol. II. Ginn And Company. pp. 284.It formed at bridge across which fools could not hope to pass, and was therefore known as the

*pons asinorum*, or bridge of fools.¹

1. The term is something applied to the Pythagorean Theorem.**^**Pons asinorum — Definition and More from the Free Merriam**^**Jaakko Hintikka, "On Creativity in Reasoning", in Ake E. Andersson, N.E. Sahlin, eds.,*The Complexity of Creativity*, 2013, ISBN 9401587884, p. 72**^**A. Battersby,*Mathematics in Management*, 1966, quoted in Deakin**^**Jeremy Bernstein, "Profiles: A.I." (interview with Marvin Minsky),*The New Yorker*December 14, 1981, p. 50-126- ^
^{a}^{b}Michael A.B. Deakin, "From Pappus to Today: The History of a Proof",*The Mathematical Gazette***74**:467:6-11 (March 1990) JSTOR 3618841 **^**Heath pp. 251–255**^**Following Proclus p. 53**^**For example F. Cuthbertson*Primer of geometry*(1876 Oxford) p. 7**^**Charles Lutwidge Dodgson,*Euclid and his Modern Rivals*Act I Scene II §6**^**Following Proclus p. 54**^**Heath p. 254 for section**^**For example J.M. Wilson*Elementary geometry*(1878 Oxford) p. 20**^**Following Wilson**^**A. M. Legendre*Éléments de géométrie*(1876 Libr. de Firmin-Didot et Cie) p. 14**^**J. R. Retherford,*Hilbert Space*, Cambridge University Press, 1993, page 27.- ^
^{a}^{b}^{c}^{d}A. F. West & H. D. Thompson "On Dulcarnon, Elefuga And Pons Asinorum as Fanciful Names For Geometrical Propositions"*The Princeton University bulletin*Vol. 3 No. 4 (1891) p. 84 **^**D.E. Smith*History of Mathematics*(1958 Dover) p. 284**^**Derbyshire, John (2003).*Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*. 500 Fifth Street, NW, Washington D.C. 20001: Joseph Henry Press. p. 202. ISBN 0-309-08549-7.first-class mathematician.

`{{cite book}}`

: CS1 maint: location (link)**^**Charles Lutwidge Dodgson,*Euclid and his Modern Rivals*Act I Scene II §1**^**W.E. Aytoun (Ed.)*The poetical works of Thomas Campbell*(1864, Little, Brown) p. 385 Google Books**^**John Stuart Mill*Principles of Political Economy*(1866: Longmans, Green, Reader, and Dyer) Book 2, Chapter 16, p. 261**^**Reid, Michael (28 October 2006). "Rubik's Cube patterns".*www.cflmath.com*. Archived from the original on 12 December 2012. Retrieved 22 September 2019.**^**Eric S. Raymond, "Why Python?",*Linux Journal,*April 30, 2000**^**Aasinsilta on laiskurin apuneuvo | Yle Uutiset | yle.fi

Look up in Wiktionary, the free dictionary.pons asinorum |

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Proposition 5 of Euclid's Elements |

- Pons asinorum at PlanetMath.
- D. E. Joyce's presentation of Euclid's
*Elements*