In an acute-angled triangle, express a median in terms of its sides.
We have
In 
ABC, AD is median
AE⊥BC
In 
AEB
AB2=AE2+BE2
AB2=AD2-DE2+(BD-DE)2
AB2=AD2-DE2+BD2-2xBDxDE+DE2
AB2=AD2+BD2-2xBDxDE
AB2=AD2+BC2/4-BCxDE …………. (I) [GIVEN BC=2BD]
In 
AEC
AC2=AE2+EC2
AC2=AD2-DE2+ (DE+CD)2
AC2=AD2-DE2+2CDxDE
AC2=AD2+BC2/4+BCxDE ……….(II) [BC=2CD]
By adding equ. (i) and (ii) we get
AB2+AC2=2AD2+BC2/2
2AB2+2AC2=4AD2+BC2 [MULTIPLY BY 2]
4AD2=2AB2+2AC2-BC2
AD2=2AB2+2AC2-BC2
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