Semiperimeter
The semiperimeter on a figure is defined as
| s=1/2p, |
(1)
|
where p 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 a, b, and c
| A_Delta=sqrt(s(s-a)(s-b)(s-c)), |
(2)
|
and Brahmagupta's formula for the area of a quadrilateral
| A_(quadrilateral)=sqrt((s-a)(s-b)(s-c)(s-d)-abcdcos^2((A+B)/2)). |
(3)
|
The semiperimeter also appears in the beautiful l'Huilier's theorem about spherical triangles.
For a triangle, the following identities hold,
Now consider the above figure. Let I be the incenter of the triangle DeltaABC, with D, E, and F the tangent points of the incircle. Extend the line BA with GA=CE. Note that the pairs of triangles (ADI,AFI), (BDI,BEI), (CFI,CEI) are congruent. Then
Furthermore,
(Dunham 1990). These equations are some of the building blocks of Heron's derivation of Heron's formula.
See also
PerimeterExplore with Wolfram|Alpha
More things to try:
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.Referenced on Wolfram|Alpha
SemiperimeterCite this as:
Weisstein, Eric W. "Semiperimeter." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Semiperimeter.html