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: / / PONZ ass-i-NOR-əm), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Euclid's Elements, and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. The term is also applied to the Pythagorean theorem.
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.
A persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem. 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.
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. 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:
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. 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.
The proof is as follows:
A standard textbook method is to construct the bisector of the angle at A. 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:
Legendre uses a similar construction in Éléments de géométrie, but taking X to be the midpoint of BC. 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. Mathematics historian William Dunham wrote that Garfield's trapezoid work was "really a very clever proof." According to the Journal, Garfield arrived at the proof "in mathematical amusements and discussions with other members of congress."
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
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:
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 sometimes applied to the Pythagorean Theorem.
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