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In aright angled triangle BAC , angle A=90 seg AD, BE, CF are medians . prove that 2(AD2+BE2CF2)=3BC2

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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 ]

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 ]