Let a,b,c be the side-lengths of a triangle, and l, m, n be the lengths of its medians. Then prove that $k =\frac{l+m+n}{a+b+c}$can assume every value in the interval $(\frac{3}{4} , 1)$ .
I can prove that $k $ belongs to $(\frac{3}{4} , 1)$. But I can not prove that $k$ assumes every value in that interval.
Can anyone please help me?
Take an isosceles triangle with sides $1$, $1$ and $x$ (with $0<x<2$). The median relative to the base $x$ has length ${1\over2}\sqrt{4-x^2}$, while the other two medians measure ${1\over2}\sqrt{1+2x^2}$ (see here for a proof). Hence: $$ k(x)={\sqrt{4-x^2}+2\sqrt{1+2x^2}\over2(2+x)}. $$ This is a continuous function in $[0,2]$ and $$ k(0)=1,\quad k(2)={3\over4}. $$ As a continuous function $k(x)$ takes then all values comprised between these, as it was to be proven.