We have
In ΔABC, AD is median
AE⊥BC
In ΔAEB
AB2 = AE2 + BE2
AB2 = AD2 - DE2 + (BD - DE)2
AB2 = AD2 - DE2 + BD2 - 2 x BD x DE + DE2
AB2 = AD2 + BD2 - 2 x BD x DE
AB2 = AD2 + BC2/4 - BC x DE …………. (I) [GIVEN BC = 2BD]
In ΔAEC
AC2 = AE2 + EC2
AC2 = AD2 - DE2 + (DE + CD)2
AC2 = AD2 - DE2 + 2CD x DE
AC2 = AD2 + BC2/4 + BC x DE ……….(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
