Mathematics - Properties of Triangle Question with Solution | TestHub

The formula for the length of a median from vertex $A$ is $m_a^2=\frac{2b^2+2c^2-a^2}{4}$. Similarly, for medians from $B$ and $C$: $m_b^2=\frac{2a^2+2c^2-b^2}{4}$ and $m_c^2=\frac{2a^2+2b^2-c^2}{4}$. Summing these three equations: $m_a^2+m_b^2+m_c^2=\frac{1}{4}[(2b^2+2c^2-a^2)+(2a^2+2c^2-b^2)+(2a^2+2b^2-c^2)]$. This simplifies to $m_a^2+m_b^2+m_c^2=\frac{1}{4}[(2-1+2+2-1)a^2+(2+2-1+2-1)b^2+(2+2-1+2-1)c^2]=\frac{1}{4}[3a^2+3b^2+3c^2]=\frac{3}{4}(a^2+b^2+c^2)$.