The semiperimeter on a figure is defined as
(1)
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where is the perimeter. The semiperimeter of polygons appears in unexpected ways in the computation of their areas. The most notable cases are in the altitude, exradius, and inradius of a triangle, the Soddy circles, Heron's formula for the area of a triangle in terms of the legs , , and
(2)
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and Brahmagupta's formula for the area of a quadrilateral
(3)
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The semiperimeter also appears in the beautiful l'Huilier's theorem about spherical triangles.
For a triangle, the following identities hold,
(4)
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(5)
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(6)
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Now consider the above figure. Let be the incenter of the triangle , with , , and the tangent points of the incircle. Extend the line with . Note that the pairs of triangles , , are congruent. Then
(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(13)
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Furthermore,
(14)
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(15)
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(16)
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(17)
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(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(Dunham 1990). These equations are some of the building blocks of Heron's derivation of Heron's formula.