Relation between sides and medians

Let the sides of the triangle be $a,b,c$ and the medians to these sides be $m_a,m_b,m_c$ respectively. We are given that, $m_a^2+m_b^2+m_c^2=51$ By the Apollonius's Theorem , we have, $\begin{aligned} a^2+b^2&=2\left(m_c^2+\left(\frac{c}{2}\right)^2\right) \\ c^2+a^2&=2\left(m_b^2+\left(\frac{b}{2}\right)^2\right) \\ b^2+c^2&=2\left(m_a^2+\left(\frac{a}{2}\right)^2\right) \\ \end{aligned}$ Adding the above equations, $\begin{aligned} 2(a^2+b^2+c^2)&=2\left(m_a^2+m_b^2+m_c^2+\frac{a^2}{4}+\frac{b^2}{4}+\frac{c^2}{4}\right) \\ (a^2+b^2+c^2)\left(1-\frac{1}{4}\right)&=m_a^2+m_b^2+m_c^2 \\ \therefore\; a^2+b^2+c^2&=\frac{4}{3}(m_a^2+m_b^2+m_c^2) \\ &=\boxed{68} \end{aligned}$