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Langley's Adventitious Angles Solution to Langley's 80-80-20 triangle problem. Langley's Adventitious Angles is a puzzle in which one must infer an angle in a geometric diagram from other given angles. It was posed by Edward Mann Langley in The Mathematical Gazette in 1922. [1] [2]
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.
Solutions to the equivalent problem of maximizing the minimum distance between n points in an isosceles right triangle, were known to be optimal for n < 8 [2] and were extended up to n = 10. [3] In 2011 a heuristic algorithm found 18 improvements on previously known optima, the smallest of which was for n = 13. [4]
The reverse triangle inequality is an equivalent alternative formulation of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is: [19] Any side of a triangle is greater than or equal to the difference between the other two sides. In the case of a normed vector space, the statement is:
The unique 6-point isosceles set in the plane. The shaded regions show four of the 20 isosceles triangles formed by triples of these points. In discrete geometry, an isosceles set is a set of points with the property that every three of them form an isosceles triangle.
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To convert between these two formulations of the problem, the square side for unit circles will be = + /. The optimal packing of 15 circles in a square Optimal solutions have been proven for n ≤ 30. Packing circles in a rectangle; Packing circles in an isosceles right triangle - good estimates are known for n < 300.
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