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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.
In geometry, an isosceles triangle (/ aɪ ˈ s ɒ s ə l iː z /) is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.
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 ...
The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof. An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.
Every triangle with two angle bisectors of equal lengths is isosceles. The theorem was first mentioned in 1840 in a letter by C. L. Lehmus to C. Sturm, in which he asked for a purely geometric proof. Sturm passed the request on to other mathematicians and Steiner was among the first to provide a solution.
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 ...
Duplicate the right triangle to form the isosceles triangle ACP. Construct the circle with center A and radius b, and its tangent h = BH through B. The tangent h forms a right angle with the radius b (Euclid's Elements: Book 3, Proposition 18; or see here), so the yellow triangle in Figure 8 is right. Apply the Pythagorean theorem to obtain
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);