What is semi perimeter

The semiperimeter on a figure is defined as

(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 

(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 of triangles , ,  are congruent. Then

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

Furthermore,

			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)