Answer :

Given:- ΔABC and AD is median

To Prove:- AB^{2} + AC^{2} = 2(AD^{2} + CD^{2})

Proof:- Let, A at origin

be position vector of B and C respectively

Therefore,

Now position vector of D, mid-point of BC i.e divides BC in 1:1.

Section formula of internal division: Theorem given below

“Let A and B be two points with position vectors

respectively, and c be a point dividing AB internally in the ration m:n. Then the position vector of c is given by

Position vector of D is given by

⇒

Now distance/length of CD

= position vector of D-position vector of C

⇒

⇒

Now taking RHS

= 2(AD^{2} + CD^{2})

=

=

=

=

=

= AB^{2} + AC^{2}

= LHS

Hence proved

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