TITLE: How to prove: $S=\frac{4}{3}\sqrt{ m(m-m_a)(m-m_b)(m-m_c)}$
QUESTION [5 upvotes]: If $$m_a, m_b, m_c$$ are the medians of a triangle and let $$m=\frac{m_a+ m_b+ m_c}{2}$$
then Area $S$ of triangle is given by
$$S=\frac{4}{3}\sqrt{ m(m-m_a)(m-m_b)(m-m_c)}$$ This looks very similar to Heron's formula. How to prove this formula?

REPLY [2 votes]: A simple way to prove this is given here without much computation: http://jwilson.coe.uga.edu/emt725/Medians.Triangle/Area.Medians.Tri.html