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
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?


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