Question

in a right angle triangle BAC angle BAC is equal to 90 degree segments AD and CF and BE are medians then prove that 2(AD square + BE square + CF square is equal to 3 BC square

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 )

Answer 2

Answer:

Step-by-step explanation:

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 )