<s>[INST] How to prove: $S=\frac{4}{3}\sqrt{ m(m-m_a)(m-m_b)(m-m_c)}$ 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?
 [/INST] $$S=\frac{4}{3}\sqrt{ m(m-m_a)(m-m_b)(m-m_c)}$$
$$ =\frac{4}{3}\sqrt{ \frac{(m_a+m_b+m_c)}{2}\frac{(-m_a+m_b+m_c)}{2}\frac{(m_a-m_b+m_c)}{2}\frac{(m_a+m_b-m_c)}{2}}$$
$$ =\frac{1}{3}\sqrt{(m_a+m_b+m_c)(-m_a+m_b+m_c)(m_a-m_b+m_c)(m_a+m_b-m_c)}$$
$$ =\frac{1}{3}\sqrt{[(m_a+m_b)^2-m_c^2] [m_c^2-(m_b-m_a)^2] } $$
$$ =\frac{1}{3}\sqrt{-[(m_a+m_b)(m_b-m_a)]^2+m_c^2(m_a^2+m_b^2)+m_c^2(m_b^2-m_a^2)-m_c^4}$$
$$ =\frac{1}{3}\sqrt{-[m_b^2-m_a^2]^2+m_c^2(m_a+m_b)^2+m_c^2(m_b-m_a)^2-m_c^2}$$
$$ =\frac{1}{3}\sqrt{2(m_a^2m_b^2+m_b^2m_c^2+m_c^2m_a^2)-(m_a^4+m_b^4+m_c^4)}$$
Now replacing $m_a$, $m_b$, $m_c$ respectively with $\frac{1}{2}\sqrt{2c^2+2b^2-a^2}$, $\frac{1}{2}\sqrt{2a^2+2c^2-b^2}$, $\frac{1}{2}\sqrt{2a^2+2b^2-c^2}$
You'll arrive at the Heron's Formula:
$$S = \sqrt{s(s-a)(s-b)(s-c)}$$
EDIT
Proof for $m_a = \frac{1}{2}\sqrt{2c^2+2b^2-a^2}$:
Assumptions: $BC = a$, $AC = b$, $AD = DE = m_a$ and $AB = c$

From cosine theorem, we have:
$$a^2 = b^2 + c^2 - 2bc*cos A$$
Now, in triangle ABE(since ACD is congruent to EBD);
$$4m_a^2 = AE^2 = b^2 + c^2 - 2bc*cos(\pi - A)$$
$$a^2 + m_a^2 = 2(b^2 + c^2) - 2[bc*cos A + bc*cos(\pi - A)]$$
or,
$$2m_a^2 = 2b^2 + 2c^2 - a^2$$
Hence,
$$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2} $$
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