Inequalities in Triangle Geometry

Theorem 1

Let \(a\), \(b\), \(c\) be the lengths of the sides of \(\triangle ABC\), \(R\), \(r\), \(r_a\), \(r_b\), \(r_c\) its circumradius, inradius, and the radii of the excircles corresponding to \(a\), \(b\), and \(c\). Denote by \(p\) the semi-perimeter of \(\triangle ABC\) and by \(S\) its area. Let \(h_a\), \(h_b\), \(h_c\) be the lengths of the altitudes, \(m_a\), \(m_b\), \(m_c\) the lengths of the medians, and \(l_a\), \(l_b\), \(l_c\) the lengths of the segments of the angle bisectors that belong to the triangle. Then the following inequalities hold: \begin{eqnarray*} 9r&\leq h_a+h_b+h_c\leq l_a+l_b+l_c\leq \sqrt{r_ar_b}+\sqrt{r< br> c}+\sqrt {r_cr_a} \leq p\sqrt 3\leq r_a+r_b+r_c = r+4R. \quad\quad\quad\quad\quad(1) \end{eqnarray*}

Problem 1 Prove that \[9r\leq h_a+h_b+h_c\leq l_a+l_b+l_c\leq m_a+m_b+m_c\leq \frac92 R.\]

Problem 2

Prove that \[27r^2\leq h_a^2+h_b^2+h_c^2\leq l_a^2+l_b^2+l_c^2\leq p^2\leq m_a^2+m_b^2+m_c^2\leq \frac{27}4R^2.\]

Problem 3

Prove that \[r\leq \frac{\sqrt{\sqrt 3 S}}{3}\leq \frac{\sqrt 3}9p\leq \frac12R.\]

Problem 4

Assume that \(M\) is the point inside the triangle \(ABC\). Let \(r\) be the inradius of the triangle. Prove that \(MA+MB+MC\geq 6r\). When does the equality hold?