p

a

b

c

m_{a}

m_{b}

m_{c}

a(p-a)

b(p-b)

c(p-c)

\left(\frac{m_{a}}{a}\right)^{2}+\left(\frac{m_{b}}{b}\right)^{2}+\left(\frac{m_{c}}{c}\right)^{2}\geqslant\frac{9}{4}

b(p-b)+c(p-c)\gt a(p-a).

b(p-b)+c(p-c)\gt a(p-a)~\Leftrightarrow~

~\Leftrightarrow~b\cdot\frac{a+c-b}{2}+c\cdot\frac{a+b-c}{2}\gt a\cdot\frac{b+c-a}{2}~\Leftrightarrow~

~\Leftrightarrow~b(a+c-b)+c(a+b-c)\gt a(b+c-a)~\Leftrightarrow~

~\Leftrightarrow~(ab+bc-b^{2})+(ac+bc-c^{2})\gt ab+ac-a^{2}~\Leftrightarrow~

~\Leftrightarrow~b^{2}+c^{2}-2bc\lt a^{2}~\Leftrightarrow~(b-c)^{2}\gt a^{2}~\Leftrightarrow~|b-c|\lt a~\Leftrightarrow~

~\Leftrightarrow~\syst{b-c\lt a\\b-c\gt-a.\\}~\Leftrightarrow~\syst{b\lt a+c\\a+b\gt c.\\}

\left(\frac{m_{a}}{a}\right)^{2}+\left(\frac{m_{b}}{b}\right)^{2}+\left(\frac{m_{c}}{c}\right)^{2}\geqslant\frac{9}{4}~\Leftrightarrow~

~\Leftrightarrow~\frac{\frac{b^{2}}{2}+\frac{c^{2}}{2}-\frac{a^{2}}{4}}{a^{2}}+\frac{\frac{a^{2}}{2}+\frac{c^{2}}{2}-\frac{b^{2}}{4}}{b^{2}}+\frac{\frac{a^{2}}{2}+\frac{b^{2}}{2}-\frac{c^{2}}{4}}{c^{2}}\geqslant\frac{9}{4}~\Leftrightarrow~

~\Leftrightarrow~\left(\frac{b^{2}}{c^{2}}+\frac{c^{2}}{a^{2}}-\frac{1}{2}\right)+\left(\frac{a^{2}}{b^{2}}+\frac{c^{2}}{b^{2}}-\frac{1}{2}\right)+\left(\frac{b^{2}}{c^{2}}+\frac{c^{2}}{a^{2}}-\frac{1}{2}\right)-\frac{3}{2}\geqslant\frac{9}{2}~\Leftrightarrow~

~\Leftrightarrow~\left(\frac{b^{2}}{a^{2}}+\frac{a^{2}}{b^{2}}\right)+\left(\frac{c^{2}}{a^{2}}+\frac{a^{2}}{c^{2}}\right)+\left(\frac{c^{2}}{b^{2}}+\frac{b^{2}}{c^{2}}\right)\geqslant6.

\frac{b^{2}}{a^{2}}+\frac{a^{2}}{b^{2}}\geqslant2,~\frac{c^{2}}{a^{2}}+\frac{a^{2}}{c^{2}}\geqslant2,~\frac{c^{2}}{b^{2}}+\frac{b^{2}}{c^{2}}\geqslant2