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for any triangle ABC prove that AB^2+AC^2=2(AD^2+BD^2) where D is the midpoint of BC.

for any triangle ABC prove that AB^2+AC^2=2(AD^2+BD^2) where D is the midpoint of BC.

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1 Answers

Ajay Verma
askIITians Faculty 33 Points
10 years ago
solution:

take the points as given in the figure


-1094_280220141570.jpg

Now LHS = AB2+ AC2
= a^2 + b^2 + (a - 2c)^2 + b^2
= a^2 + b^2 + a^2 - 4ac + 4*c^2 + b^2
= 2 a^2 + 2 b^2 + 4 c^2 - 4 ac ......................................................... (1)


RHS = 2 (AD^2 + BD^2)

= 2 [ (a - c)^2 + b^2 + c^2 ]
= 2 [ a^2 - 2 ac + c^2 + b^2 + c^2]
= 2 [ a^2 + b^2 + 2 c^2 - 2 ac]
= 2 a^2 + 2 b^2 + 4 c^2 - 4 ac ......................................................... (2)

eqn (1 ) = eqn(2)
so LHS = RHS



Thanks and Regards,
Ajay verma,
askIITians faculty,
IIT HYDERABAD

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