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)

(23)

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