# 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?