Exploring a Triangle - Medians, Altitudes and Angle Bisectors!

Geometry Level 5

Let \(m_a, h_a, w_a\) denote the lengths of the median, the altitude and the internal angle bisector, respectively, to side \(A\) in a \(\Delta ABC\). Define \(m_b, m_c, h_b, h_c, w_b, w_c\) similarly. Let \(R\) be the circumradius of \(\Delta ABC\). Let:

  • \(\eta_1 = \max \left( \displaystyle \dfrac{1}{R} \sum_{cyclic} \dfrac{b^2+c^2}{m_a} \right) \)

  • \(\eta_2 = \min \left( \displaystyle \dfrac{1}{R} \sum_{cyclic} \dfrac{b^2+c^2}{h_a} \right) \)

  • \(\eta_3 = \min \left( \displaystyle \dfrac{1}{R} \sum_{cyclic} \dfrac{b^2+c^2}{w_a} \right) \)

Find the value of \(\large{ \left \lfloor \eta_1 + \eta_2 + \eta_3 \right \rfloor} \).


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