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 ]