triangle inequality given perimeter and area

Question

Show that the following inequality holds between the perimeter $p$ and the area of the triangle $a$.

$$p^2 \ge 12\sqrt3\ a$$

Answer 1

Using Heron's formula, we can write the desired inequality as: $$ (2s)^2 \ge 12 \sqrt3 \sqrt{s(s-a)(s-b)(s-c)} $$

which on squaring, simplifies to $$ s^3 \ge 27 (s-a)(s-b)(s-c) $$

Can you recognise the AM-GM now?