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In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . Letting be the semiperimeter of the triangle, = (+ +), the area is [1]
Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides a , b , and c respectively as m a , m b , and m c and their semi-sum ( m a + m b + m c )/2 as σ, we have [ 10 ]
In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area A are all positive integers. [ 1 ] [ 2 ] Heronian triangles are named after Heron of Alexandria , based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides 13, 14, 15 and area 84 .
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter s.
A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula. If the semiperimeter is not used, Brahmagupta's formula is
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Euclid's construction for proof of the triangle inequality for plane geometry. Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. [6] Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB.