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. There are two possible explanations for the name pons asinorum, the simplest being that the diagram used resembles an actual bridge. Though this etymology is dubious, it is echoed in Chaucer's use of the term 'flemyng of wreches' for the theorem.

Where Оё 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.ДЄnother 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'.

Draw the lines BE, DC and DE.Д¬onsider the triangles BAE and CAD BAВ = CA, AEВ = AD, and в€ A Pick an arbitrary point D on side AB and construct E on AC so that ADВ = AE. Let ABC be an isosceles triangle with AB and AC being the equal sides. Proclus' variation of Euclid's proof proceeds as follows: The proof relies heavily on what is today called side-angle-side, the previous proposition in the Elements. 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. 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. 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. Euclid's proof involves drawing auxiliary lines to these extensions. Д®uclid's Elements Book 1 proposition 5 the pons asinorumД®uclid'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. 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. ДЄ persistent piece of mathematical folklore claims that an artificial intelligence program discovered an original and more elegant proof of this theorem. Its first known usage in this was in 1645. 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. The term is also applied to the Pythagorean theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. This statement is Proposition 5 of Book 1 in Euclid's Elements, and is also known as the isosceles triangle theorem. 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:, English: / Л€ p Й’ n z ЛЊ Г¦ s ЙЄ Л€ n Й”Лђr Й™ m/ PONZ ass-i- NOR-Й™m), typically translated as 'bridge of asses'. It is pronounced eye- sos-uh-lees.The pons asinorum in Byrne's edition of the Elements showing part of Euclid's proof. Isosceles comes from the Greek word isoskeles, itself a compound word, from iso, meaning 'same', and skelos, meaning 'leg'. This is the only right triangle that is also an isosceles triangle. It has one right angle (90В°), and two 45В° angles. The right isosceles triangle (AKA the 45-45-90 triangle) is one of the most commonly seen triangles, and is used widely in architecture and engineering.

The vertex will be centered directly over the midpoint of the base. This puts the odd angle out at the top, making it the vertex angle. When only two sides are equal, it is traditional to refer to the unequal side as the base of the triangle. These are in contrast to scalene triangles, in which all sides are different lengths. This will mean that it also has two congruent (identical) angles, and it is bilaterally symmetrical.ДЄn equilateral triangle is a special case of the isosceles triangle, in which all three sides are the same length, and all of its internal angles are also the same, 60В°. An isosceles triangle is a triangle with at least two congruent sides that is to say, at least two sides are exactly the same length.

Isosceles triangles