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 pronunciation: [ˈpons asiˈnoːrʊm]; English /ˈpɒnz ˌæsᵻˈnɔərəm/ PONZ-ass-i-NOR-(r)əm), Latin for "bridge of donkeys". 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 name of this statement is also used metaphorically for a problem or challenge which will separate the sure of mind from the simple, the fleet thinker from the slow, the determined from the dallier; to represent a critical test of ability or understanding.'

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, given that it makes the proof more complicated, Euclid added the second conclusion to the theorem. 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.

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