Let denote the area of a triangle with sides of lengths . Heron’s formula states that

We prove this formula using vectors via the following steps. Assume the triangle has vertices , and with

- Using the identities
prove the formula

- Simplify the formula in part (a) to obtain the formula
and use this to deduce Heron’s formula.

*Proof.*We know the area of the triangle (in terms of the vectors and ) isTherefore,

*Proof.*We simplify the formula in part (a),