In triangle ABC if AD is the median then show that AB² + AC² = 2(AD² + BD²)

in triangle ABC, AD is median 

then , BD = BC 

in triangles ABD and ADC

AB² = AD² + BD^{2 .........(1)}

AC² = AD² + BC^{2 ............(2)}

now add both the equations,

AB² + AC² = AD² + AD² + BD²+ BC²

AB² + AC² = 2AD² + 2BD²  [ since BD = BC]

AB² + AC² = 2[AD² + BD²]

hence proved...

HOPE THIS WOULD HELP YOU OUT