Question

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

Answer 1

see explanation

Explanation:

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

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)