proof of Heron’s formula

Let α be the angle between the sides b and c, then we get from the cosines law:


Using the equation


we get:


Now we know:


So we get:

Δ = 14-a4-b4-c4+2b2c2+2a2b2+2a2c2
= 14(a+b+c)(b+c-a)(a+c-b)(a+b-c)
= s(s-a)(s-b)(s-c).

This is Heron’s formula.

Title proof of Heron’s formula
Canonical name ProofOfHeronsFormula
Date of creation 2013-03-22 12:41:38
Last modified on 2013-03-22 12:41:38
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 5
Author mathwizard (128)
Entry type Proof
Classification msc 51-00