The semiperimeter on a figure is defined as
![\begin{displaymath}
s\equiv {\textstyle{1\over 2}}p,
\end{displaymath}](s1_664.gif) |
(1) |
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
![\begin{displaymath}
A_\Delta=\sqrt{s(s-a)(s-b)(s-c)},
\end{displaymath}](s1_665.gif) |
(2) |
and Brahmagupta's Formula for the Area of a Quadrilateral
![\begin{displaymath}
A_{\rm quadrilateral}=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left({A+B\over 2}\right)}.
\end{displaymath}](s1_666.gif) |
(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
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
Furthermore,
(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.
© 1996-9 Eric W. Weisstein
1999-05-26