Apollonius's theorem gives the relationship between the length of the median of a triangle to the lengths of its sides. It states that 

"The sum of the squares of any two sides of any triangle equals twice the sum of the square of half the third side and square on the median bisecting the third side". 

Proof of Apollonius' Theorem

Apollonius' Theorem can be proved using Pythagorean Theorem. Let \(BD = DC = m\) and \(ED = n\) 

In the triangle, \(AEB\), \[A{B^2} = {(m - n)^2} + A{E^2}\] \[A{C^2} = {(m + n)^2} + A{E^2}\] Adding the two equations, \[A{B^2} + A{C^2} = 2({m^2} + {n^2}) + 2(A{E^2})\]But \(A{E^2} = A{D^2} - {n^2}\),  putting the value of \(A{E^2}\), we get \[A{B^2} + A{C^2} = 2({m^2} + {n^2}) + 2(A{D^2} - {n^2})\]

\[\bbox[5px, border: 2px solid #0071dc]{\color{blue}{A{B^2} + A{C^2} = {\rm{ }}2{\rm{ }}\left( {A{D^2} + D{C^2}} \right)}}\]