Pons asinorum
In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ˈpɒnz ˌæsɪˈnɔːrəm/ PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem.
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.
[2] 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.
[3] 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.
[4] 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.
[4] The name pons asinorum has itself occasionally been applied to the Pythagorean theorem.
[5] 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.
[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 (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:[8] Let
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 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.
[11] The proof is as follows:[12] Let ABC be an isosceles triangle with AB and AC being the equal sides.
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 be ABC with AB = AC.
and extend it to meet BC at X. AB = AC and AX is equal to itself.
Legendre uses a similar construction in Éléments de géométrie, but taking X to be the midpoint of BC.
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.
[17] Mathematics historian William Dunham wrote that Garfield's trapezoid work was "really a very clever proof.
"[18] According to the Journal, Garfield arrived at the proof "in mathematical amusements and discussions with other members of congress.
"[19] The isosceles triangle theorem holds in inner product spaces over the real or complex numbers.
Uses of the pons asinorum as a metaphor for a test of critical thinking include: A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem.
[22][23] 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.
