Given:- ΔABC and AD is median

To Prove:- AB² + AC² = 2(AD² + CD²)

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 formulae 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(AD² + CD²)

=

=

=

=

=

= AB² + AC²

= LHS

Hence proved