302 The area of a triangle
9.1 Heron’s formula for the area of a triangle
Theorem 9.1.The area of a triangle of sidelengthsa,b,cis given by
=√
s(s−a)(s−b)(s−c),wheres=^12 (a+b+c).
IaY′ YIABCrras−b s−c s−aProof.Consider the incircle and the excircle on the opposite side ofA.
From the similarity of trianglesAIZandAI′Z′,
r
ra=
s−a
s.
From the similarity of trianglesCIYandI′CY′,
r·ra=(s−b)(s−c).From these,
r=√
(s−a)(s−b)(s−c)
s,
and the area of the triangle is
=rs=√
s(s−a)(s−b)(s−c).