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The pons asinorum in Oliver Byrne's edition of the Elements [1]. In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ ˈ p ɒ n z ˌ æ s ɪ ˈ n ɔːr ə m / PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem.
A direct proof using classical geometry was developed by James Mercer in 1923. [2] This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles.
Lexell's proof by breaking the triangle A ∗ B ∗ C into three isosceles triangles. The main idea in Lexell's c. 1777 geometric proof – also adopted by Eugène Catalan (1843), Robert Allardice (1883), Jacques Hadamard (1901), Antoine Gob (1922), and Hiroshi Maehara (1999) – is to split the triangle into three isosceles triangles with common apex at the circumcenter and then chase angles ...
An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point B: [10] (i) as depicted (which is to be proved), or (ii) B coincident with D (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle γ, which would violate the triangle postulate), or lastly ...
Since OA = OB = OC, OBA and OBC are isosceles triangles, and by the equality of the base angles of an isosceles triangle, ∠ OBC = ∠ OCB and ∠ OBA = ∠ OAB. Let α = ∠ BAO and β = ∠ OBC. The three internal angles of the ∆ABC triangle are α, (α + β), and β. Since the sum of the angles of a triangle is equal to 180°, we have
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);