In A B C the mid points of the sides A B, B C and C A are respectively (l, 0,0),(0, m, 0) and (0,0, n) . Then, A B^2+B C^2+C A^2 l^2+m^2+n^2 is equal to

Step-by-step solution

From the figure, gathered x₁+x₂=2 l, y₁+y₂=0, z₁+z₂=0, x₂+x₃=0, y₂+y₃=2 m, z₂+z₃=0 and x₁+x₃=0, y₁+y₃=0, z₁+z₃=2 n gathered On solving, we get aligned x₁ & =l, x₂=l, x₃=-l, & y₁=-m, y₂=m, y₃=m and z₁ & =n, z₂=-n, z₃=n aligned Coordinates are A(l,-m, n), B(l, m,-n) and C(-l, m, n) aligned & A B^2+B C^2+C A^2 l^2+m^2+n^2 = & (4 m^2+4 n^2 )+ (4 l^2+4 n^2 )+ (4 l^2+4 m^2 ) l^2+m^2+n^2 = & 8 aligned