A triangle with sides of lengths a, b, and c has area A given by where is the semiperimete...

A triangle with sides of lengths a, b, and c has area A given by

where

is the semiperimeter of the triangle. This is known as Heron’s formula.4 Prove Heron’s formula. (Hint: If two sides of the triangle are regarded as vectors a and b then ; use Exercise 1.)

4Heron of Alexandria, circa first century A.D., is thought to have been an Egyptian influenced by Greek thought. He wrote widely on geometry and mechanics.

Exercise 1

Prove that for any four vectors a, b, c, and d in ℝ3,

 (a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c).