For any A B C , if the median through A is m₁ , the median through B is m₂ , the median through C is m₃ and [ array l a^2 b^2 c^2 array ]=M [ array l m₁^2 m₂^2 m₃^2 array ] for a certain 3 3 matrix M . Then:

Step-by-step solution

m₁^2= 2 b^2+2 c^2-a^2 4 ; m₂^2= 2 c^2+2 a^2-b^2 4 ; m₃^2= 2 a^2+2 b^2-c^2 4 array ll & [ array l m₁^2 m₂^2 m₃^2 array ]= 1 4 [ array ccc -1 & 2 & 2 2 & -1 & 2 2 & 2 & -1 array ] [ array l a^2 b^2 c^2 array ] & [ array l m₁^2 m₂^2 m₃^2 array ]= A 4 [ array l a^2 b^2 c^2 array ] 4 A⁻¹ [ array l m₁^2 m₂^2 m₃^2 array ]= [ array l a^2 b^2 c^2 array ] array array ll & [ array l a^2 b^2 c^2 array ]= 4 9 [ array ccc -1 & 2 & 2 2 & -1 & 2 2 & 2 & -1 array ] [ array l m₁^2 m₂^2 m₃^2 array ] & M= 4 9 [ array ccc -1 & 2 & 2 2