Ingeometry, thetheorem that theangles opposite the equal sides of anisosceles triangle are themselves equal is known as thepons asinorum (/ˈpɒnzˌæsɪˈnɔːrəm/PONZ ass-ih-NOR-əm), Latin for "bridge ofasses", or more descriptively as theisosceles triangle theorem. The theorem appears as Proposition 5 of Book 1 inEuclid'sElements.^[1] Itsconverse is also true: if two angles of atriangle are equal, then the sides opposite them are also equal.

Pons asinorum is also usedmetaphorically for a problem or challenge which acts as a test ofcritical thinking, referring to the "asses' bridge's" ability to separate capable and incapable reasoners. Its first known usage in this context was in 1645.^[2]

There are two common explanations for the namepons asinorum, the simplest being that the diagram used resembles a physicalbridge. But the more popular explanation is that it is the first real test in theElements of the intelligence of the reader and functions as a "bridge" to the harder propositions that follow.^[3]

Another medieval term for the isosceles triangle theorem wasElefuga which, according toRoger Bacon, comes from Greekelegia "misery", and Latinfuga "flight", that is "flight of the wretches". Though this etymology is dubious, it is echoed inChaucer's use of the term "flemyng of wreches" for the theorem.^[4]

The nameDulcarnon was given to the 47th proposition of Book I of Euclid, better known as thePythagorean theorem, after the ArabicDhū '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. That term has similarly been used as a metaphor for a dilemma.^[4] The namepons asinorum has itself occasionally been applied to the Pythagorean theorem.^[5]

Carl Friedrich Gauss supposedly once suggested that understandingEuler's identity might play a similar role, as a benchmark indicating whether someone could become a first-classmathematician.^[6]

Euclid's statement of thepons 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'sproof involves drawing auxiliary lines to these extensions. But, as Euclid's commentatorProclus 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 calledside-angle-side (SAS), the previous proposition in theElements, which says that given two triangles for which two pairs of corresponding sides and their included angles are respectivelycongruent, then the triangles are congruent.

Proclus' variation of Euclid's proof proceeds as follows:^[8] Let⁠${\displaystyle \mathrm{△}ABC}$⁠ be an isosceles triangle with congruent sides⁠${\displaystyle AB\cong AC}$⁠. Pick an arbitrary point⁠${\displaystyle D}$⁠ on side⁠${\displaystyle AB}$⁠ and then construct point⁠${\displaystyle E}$⁠ on⁠${\displaystyle AC}$⁠ to make congruent segments⁠${\displaystyle AD\cong AE}$⁠. Draw auxiliary line segments⁠${\displaystyle BE}$⁠,⁠${\displaystyle DC}$⁠, and⁠${\displaystyle DE}$⁠. By side-angle-side, the triangles⁠${\displaystyle \mathrm{△}BAE\cong \mathrm{△}CAD}$⁠. Therefore⁠${\displaystyle \mathrm{\angle }ABE\cong \mathrm{\angle }ACD}$⁠,⁠${\displaystyle \mathrm{\angle }ADC\cong \mathrm{\angle }AEB}$⁠, and⁠${\displaystyle BE\cong CD}$⁠. By subtracting congruent line segments,⁠${\displaystyle BD\cong CE}$⁠. This sets up another pair of congruent triangles,⁠${\displaystyle \mathrm{△}DBE\cong \mathrm{△}ECD}$⁠, again by side-angle-side. Therefore⁠${\displaystyle \mathrm{\angle }BDE\cong \mathrm{\angle }CED}$⁠ and⁠${\displaystyle \mathrm{\angle }BED\cong \mathrm{\angle }CDE}$⁠. By subtracting congruent angles,⁠${\displaystyle \mathrm{\angle }BDC\cong \mathrm{\angle }CEB}$⁠. Finally⁠${\displaystyle \mathrm{△}BDC\cong \mathrm{△}CEB}$⁠ by a third application of side-angle-side. Therefore⁠${\displaystyle \mathrm{\angle }CBD\cong \mathrm{\angle }BCE}$⁠, which was to be proved.

Proclus gives a much shorter proof attributed toPappus 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][10]This method is lampooned byCharles Dodgson inEuclid and his Modern Rivals, calling it an "Irish bull" because it apparently requires the triangle to be in two places at once.^[11]

The proof is as follows:^[12] LetABC be an isosceles triangle withAB andAC being the equal sides. Consider the trianglesABC andACB, whereACB is considered a second triangle with verticesA,C andB corresponding respectively toA,B andC in the original triangle.${\displaystyle \mathrm{\angle }A}$ is equal to itself,AB = AC andAC = AB, so by side-angle-side, trianglesABC andACB are congruent. In particular,${\displaystyle \mathrm{\angle }B=\mathrm{\angle }C}$.^[13]

A standard textbook method is to construct thebisector of the angle atA.^[14] 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:^[15] As before, let the triangle beABC withAB = AC. Construct the angle bisector of${\displaystyle \mathrm{\angle }BAC}$ and extend it to meetBC atX.AB = AC andAX is equal to itself. Furthermore,${\displaystyle \mathrm{\angle }BAX=\mathrm{\angle }CAX}$, so, applying side-angle-side, triangleBAX and triangleCAX are congruent. It follows that the angles atB andC are equal.

Legendre uses a similar construction inÉléments de géométrie, but takingX to be the midpoint ofBC.^[16] The proof is similar butside-side-side must be used instead of side-angle-side, and side-side-side is not given by Euclid until later in theElements.

The isosceles triangle theorem holds ininner product spaces over thereal orcomplex numbers. In such spaces, given vectorsx,y, andz, the theorem says that if${\displaystyle x+y+z=0}$ and${\displaystyle \Vert x\Vert =\Vert y\Vert ,}$ then${\displaystyle \Vert x-z\Vert =\Vert y-z\Vert .}$^[17]

Since${\displaystyle \Vert x-z{\Vert }^2=\Vert x{\Vert }^2-2x\cdot z+\Vert z{\Vert }^2}$ and${\displaystyle x\cdot z=\Vert x\Vert \Vert z\Vert \cos \theta ,}$ whereθ 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 thepons asinorum as a metaphor for a test of critical thinking include:

• Richard Aungerville's 14th centuryThe 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.^[4]
• The termpons asinorum, in both its meanings as a bridge and as a test, is used as a metaphor for finding the middle term of asyllogism.^[4]
• The 18th-century poetThomas 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.^[18]
• EconomistJohn Stuart Mill calledRicardo'slaw of rent thepons asinorum of economics.^[19]
• TheFinnishaasinsilta andSwedishåsnebrygga is a literary technique where a tenuous, even contrived connection between two arguments or topics, which is almost but not quite anon sequitur, is used as an awkward transition between them. In serious text, it is considered a stylistic error, since it belongs properly to thestream of consciousness- orcauserie-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").
• InDutch,ezelsbruggetje ('little bridge of asses') is the word for amnemonic. The same is true for theGermanEselsbrücke.
• InCzech,oslí můstek has two meanings – it can describe either a contrived connection between two topics or a mnemonic.

A persistent piece of mathematical folklore claims that anartificial intelligence program discovered an original and more elegant proof of this theorem.^[20][21] In fact,Marvin Minsky recounts that he had rediscovered thePappus proof (which he was not aware of) by simulating what a mechanical theorem prover might do.^[22][10]

• Euclid, commentary and trans. byT. L. HeathElements Vol. 1 (1908 Cambridge)Google Books
• Euclid, commentary byProclus, ed. and trans. byT. TaylorElements Vol. 2 (1789)Google Books

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Proposition 5 of Euclid's Elements

• Pons asinorum atPlanetMath.
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