“Semiperimeter”的概念、定义、习题解题方法及翻译-数学辞典

单词

Semiperimeter

释义

Semiperimeter

The semiperimeter on a figure is defined as

(1)

where

is the Perimeter. The semiperimeter of Polygons appears in unexpected ways in thecomputation of their Areas. The most notable cases are in the Altitude, Exradius, andInradius of a Triangle, the Soddy Circles, Heron's Formula for the Area of aTriangle in terms of the legs

, and

(2)

and Brahmagupta's Formula for the Area of a Quadrilateral

(3)

The semiperimeter also appears in the beautiful L'Huilier's Theorem about Spherical Triangles.

For a Triangle, the following identities hold,

			(4)
			(5)
			(6)

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 oftriangles

are congruent. Then





			(7)

Furthermore,



			(8)


			(9)
			(10)

(Dunham 1990). These equations are some of the building blocks of Heron's

derivation of Heron's Formula.

References

Dunham, W. ``Heron's Formula for Triangular Area.'' Ch. 5 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 113-132, 1990.

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