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