{{Short description|Geometric theorem about isosceles triangles}} {{italic title}} [[File:Byrne_pons_asinorum.jpg|right|thumb|upright=1.25|The ''pons asinorum'' in Oliver Byrne's edition of the ''Elements''<ref name=":0">{{Cite book |last=Byrne |first=Oliver |title=The First Six Books of The Elements of Euclid in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners |publisher=Taschen |year=1847 |isbn=978-1528770439 |pages=Page 5 |language=English}}</ref>]]

In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the '''''pons asinorum''''' ({{IPAc-en|ˈ|p|ɒ|n|z|_|ˌ|ae|s|ᵻ|ˈ|n|ɔər|ə|m}} {{respell|PONZ|_|ass|ih|NOR|əm}}), Latin for "bridge of asses", or more descriptively as the '''isosceles triangle theorem'''. The theorem appears as Proposition 5 of Book 1 in Euclid's ''Elements''.<ref name=":0" /> Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal.

{{anchor|metaphor1}}''Pons asinorum'' is also used metaphorically for a problem or challenge which acts as a test of critical thinking, referring to the "asses' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645.<ref>{{cite encyclopedia |title=Pons asinorum |url=http://www.merriam-webster.com/dictionary/pons%20asinorum |encyclopedia=Merriam-Webster.com Dictionary }}</ref>

== Etymology ==

There are two common explanations for the name ''pons asinorum'', the simplest being that the diagram used resembles a physical 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.<ref>D.E. Smith ''History of Mathematics'' (1958 Dover) p. 284</ref>

Another medieval term for the isosceles triangle theorem 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.<ref name="PUb"/>

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-Qarnayn'' {{lang|ar|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. That term has similarly been used as a metaphor for a dilemma.<ref name="PUb">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</ref> The name ''pons asinorum'' has itself occasionally been applied to the Pythagorean theorem.<ref>{{Cite book |title=History Of Mathematics |last=Smith |first=David Eugene |publisher=Ginn & Co. |year=1925 |at={{pgs|284}}, footnote 1 |url=https://archive.org/details/historyofmathema031897mbp/page/n299 |volume=2 }}</ref>

Carl Friedrich Gauss supposedly once suggested that understanding Euler's identity might play a similar role, as a benchmark indicating whether someone could become a first-class mathematician.<ref name=First-Class>{{cite book|last=Derbyshire|first=John|title=Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics|year=2003|publisher=Joseph Henry Press|location=500 Fifth Street, NW, Washington D.C. 20001|isbn=0-309-08549-7|page=[https://archive.org/details/primeobsessionbe00derb_0/page/202 202]|url=https://archive.org/details/primeobsessionbe00derb_0|url-access=registration|quote=first-class mathematician.}}</ref>

== Proofs ==

=== Euclid and Proclus === {| style="float:right" | right|thumb|upright=0.8|Proclus' proof |} {| style="float:right" | right|thumb|upright=0.8|''Elements'' I.5, the ''pons asinorum'' |}

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.<ref>Heath pp. 251–255</ref> The proof relies heavily on what is today called side-angle-side (SAS), the previous proposition in the ''Elements'', which says that given two triangles for which two pairs of corresponding sides and their included angles are respectively congruent, then the triangles are congruent.

Proclus' variation of Euclid's proof proceeds as follows:<ref>Following Proclus p. 53</ref> Let {{tmath|\triangle ABC }} be an isosceles triangle with congruent sides {{tmath|AB \cong AC}}. Pick an arbitrary point {{tmath|D}} on side {{tmath|AB}} and then construct point {{tmath|E}} on {{tmath|AC}} to make congruent segments {{tmath|AD \cong AE}}. Draw auxiliary line segments {{tmath|BE}}, {{tmath|DC}}, and {{tmath|DE}}. By side-angle-side, the triangles {{tmath| \triangle BAE \cong \triangle CAD}}. Therefore {{tmath|\angle ABE \cong \angle ACD}}, {{tmath|\angle ADC \cong \angle AEB}}, and {{tmath|BE \cong CD}}. By subtracting congruent line segments, {{tmath|BD \cong CE}}. This sets up another pair of congruent triangles, {{tmath|\triangle DBE \cong \triangle ECD}}, again by side-angle-side. Therefore {{tmath|\angle BDE \cong \angle CED}} and {{tmath|\angle BED \cong \angle CDE}}. By subtracting congruent angles, {{tmath|\angle BDC \cong \angle CEB}}. Finally {{tmath|\triangle BDC \cong \triangle CEB}} by a third application of side-angle-side. Therefore {{tmath|\angle CBD \cong \angle BCE}}, which was to be proved.

=== Pappus ===

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.<ref>For example F. Cuthbertson ''Primer of geometry'' (1876 Oxford) p. 7</ref><ref name="deakin"/> This method is lampooned by Charles 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.<ref>Charles Lutwidge Dodgson, ''Euclid and his Modern Rivals'' Act I Scene II §6</ref>

The proof is as follows:<ref>Following Proclus p. 54</ref> 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. <math>\angle A</math> is equal to itself, ''AB''&nbsp;=&nbsp;''AC'' and ''AC''&nbsp;=&nbsp;''AB'', so by side-angle-side, triangles ''ABC'' and ''ACB'' are congruent. In particular, <math>\angle B = \angle C</math>.<ref>Heath p. 254 for section</ref>

=== Others === right|thumb|195px|A textbook proof

A standard textbook method is to construct the bisector of the angle at ''A''.<ref>For example J.M. Wilson ''Elementary geometry'' (1878 Oxford) p. 20</ref> 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 Euclid's propositions would have to be changed to avoid the possibility of circular reasoning.

The proof proceeds as follows:<ref>Following Wilson</ref> As before, let the triangle be ''ABC'' with ''AB''&nbsp;=&nbsp;''AC''. Construct the angle bisector of <math>\angle BAC</math> and extend it to meet ''BC'' at ''X''. ''AB''&nbsp;=&nbsp;''AC'' and ''AX'' is equal to itself. Furthermore, <math>\angle BAX = \angle CAX</math>, 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''.<ref>A. M. Legendre ''Éléments de géométrie'' (1876 Libr. de Firmin-Didot et Cie) p. 14</ref> 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 inner product spaces == The isosceles triangle theorem holds in inner product spaces over the real or complex numbers. In such spaces, given vectors ''x'', ''y'', and ''z'', the theorem says that if <math>x + y + z = 0</math> and <math>\|x\| = \|y\|,</math> then <math>\|x - z\| = \|y - z\|.</math><ref>{{Cite book |last=Retherford |first=James Ron |title=Hilbert space: compact operators and the trace theorem |date=1993 |publisher=Cambridge university press |isbn=978-0-521-41884-3 |series=London mathematical society student texts |location=Cambridge|page=[https://books.google.com/books?id=IEixfs1Q514C&pg=PA27 27]}}</ref>

Since <math>\|x - z\|^2 = \|x\|^2 - 2x\cdot z + \|z\|^2</math> and <math>x \cdot z = \|x\|\|z\|\cos\theta,</math> 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.

== Metaphorical usage ==

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

* Richard Aungerville's 14th century The 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.<ref name="PUb"/> * 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.<ref name="PUb"/> * 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.<ref>{{Cite book |last=Campbell |first=Thomas |url=https://books.google.com/books?id=vFMOAAAAIAAJ |title=The Poetical Works of Thomas Campbell |date=1864 |publisher=Little, Brown |language=en}}</ref> * Economist John Stuart Mill called Ricardo's law of rent the ''pons asinorum'' of economics.<ref>John Stuart Mill ''Principles of Political Economy'' (1866: Longmans, Green, Reader, and Dyer) Book 2, Chapter 16, p. 261</ref> * 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. 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.

== Artificial intelligence proof myth ==

A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem.<ref>Jaakko Hintikka, "On Creativity in Reasoning", in Ake E. Andersson, N.E. Sahlin, eds., ''The Complexity of Creativity'', 2013, {{isbn|9401587884}}, p. 72</ref><ref>A. Battersby, ''Mathematics in Management'', 1966, quoted in Deakin</ref> 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.<ref>Jeremy Bernstein, "Profiles: A.I." (interview with Marvin Minsky), ''The New Yorker'' December 14, 1981, p. 50-126</ref><ref name="deakin">Michael A.B. Deakin, "From Pappus to Today: The History of a Proof", ''The Mathematical Gazette'' '''74''':467:6-11 (March 1990) {{JSTOR|3618841}}</ref>

== Notes == {{reflist |25em }}

== References == * Euclid, commentary and trans. by T. L. Heath ''Elements'' Vol. 1 (1908 Cambridge) [https://books.google.com/books?id=UhgPAAAAIAAJ Google Books] * Euclid, commentary by Proclus, ed. and trans. by T. Taylor ''Elements'' Vol. 2 (1789) [https://books.google.com/books?id=2SBeEnT45gsC Google Books]

== External links == {{wiktionary}}{{wikisource|Page:The_Elements_of_Euclid_for_the_Use_of_Schools_and_Colleges_-_1872.djvu/34|Proposition 5 of Euclid's Elements}} *{{PlanetMath|urlname=PonsAsinorum|title=Pons asinorum}} *[http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html D. E. Joyce's presentation of Euclid's ''Elements'']

{{Ancient Greek mathematics}}

Category:History of mathematics Category:Elementary geometry Category:Latin words and phrases Category:Articles containing proofs Category:Euclidean geometry Category:Theorems about special triangles