Question

in right angled triangle BAC,angle BAC is 90 segment AD,BE and CF are medians . Prove 2[AD2+BE2 +CF2]=3BC2

Answer 1

Draw right angle triangle ABC,

draw medians.  AD, BE, CF.  Now Join DF.
Since D and F are midpoints of sides AB and BC,  DF will be parallel to AC and is equal to 1/2 AC.

ADF, ABE, AFC are all  right angle triangles.

LHS = 2 (AD² +  BE² +  CF² )
        = 2 [ (AF² + DF²) + (AB² + AE²) + (AF² + AC²) ]
     =  2 [ (AB²/4 + AC²/4)  + (AB² + AC²) + (AC²/4 + AB²/4) ]
       = 2 [ BC² /2 +  BC² ]
       = 3 ( BC² ]