The semiperimeter on a figure is defined as
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
and Brahmagupta's formula for the area
of a quadrilateral
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.