a nice trigonometric ineq

nttu

Prove that :
$\frac{m_a.cosA}{sinB.sinC} + \frac{m_b.cosB}{sinA.sinC} + \frac{m_c.cosc}{sinB.sinA} \geq 3R$

**Posted:** Thu Aug 04, 2005 8:43 am

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Nguyen Tuan Tu

Rushil

What are the $m_{a}$ ...???? Sad

**Posted:** Thu Aug 04, 2005 3:49 pm

nttu

$m_a, m_b , m_c$ are the medians of the triangle $ABC$

**Posted:** Thu Aug 04, 2005 6:28 pm

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Nguyen Tuan Tu

Virgil Nicula

Your inequality is equivalently with $\sum am_a\cos A\ge 3S\ or\ \sum \frac{\cos A}{\sin {\phi_a}}\ge \frac 32$ , where $\phi_a,\ \phi_b,\ \phi_c$ are the measures of the angles between the median and the side coresponding to its.

We know: $\sum \cos A=1+\frac rR;\ 6sr\le \sum am_a \le 2s(R+r);\ a.s.o.$

**Posted:** Thu Aug 04, 2005 11:27 pm