Prove thatsum of any two sides of a triangle is greater than twice the medians with respect to third side ?

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Sep 22, 2017

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see explanation

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enter image source here
See Fig 1, Let #D# be the midpoint of #BC#,
#=> AD# is a median of #DeltaABC#
Now we need to prove #AB+AC>2AD#

enter image source here
See Fig2, extend #AD# to #E# such that #AD=DE#
#=> color(red)(AE=2AD)#,
Draw lines #BE and EC#, as shown in the figure.
#=> ABEC# is a parallelogram.
#=> color(red)(BE=AC)#.
Consider #DeltaABE#
Recall that the sum of any two sides of a triangle is greater than the length of the third side,
#=> AB+BE>AE#
#=> AB+AC>2AD# ....... (proved)

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