Question 34
Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a = |BC|$, $b = |CA|$, $c = |AB|$. Denote by $m_a$, $m_b$, $m_c$, respectively, the lengths of the medians which connect $A$, $B$, $C$, respectively, with the centres of the corresponding opposite sides.
a. Prove that $2m_a \lt b + c$. Deduce that $m_a + m_b + m_c \lt a + b + c$.
b. Give an example of
i. a triangle in which $m_a \gt \sqrt{bc}$;
ii. a triangle in which $m_a \le \sqrt{bc}$.
a. Prove that $2m_a \lt b + c$. Deduce that $m_a + m_b + m_c \lt a + b + c$.
b. Give an example of
i. a triangle in which $m_a \gt \sqrt{bc}$;
ii. a triangle in which $m_a \le \sqrt{bc}$.