# triangle inequality given perimeter and area

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

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

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## 1 Answer

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?

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