Given:- Quadrilateral ABCD with AC and BD are diagonals. P and Q are mid-point of AC and BD respectively

To Prove:- AB² + BC² + CD² + DA² = AC² + BD² + 4PQ²

Proof:- Let, O at Origin

be position vector of A, B, C and D respectively

As P and Q are mid-point of AC and BD,

Then, position vector of P, mid-point of AC i.e divides AC in 1:1

and position vector of Q, mid-point of BD i.e divides BD 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 “

Hence

Position vector of P is given by

Position vector of Q is given by

Distance/length of PQ

⇒

⇒

Distance/length of AC

⇒

⇒

Distance/length of BD

⇒

⇒

Distance/length of AB

⇒

⇒

Distance/length of BC

⇒

⇒

Distance/length of CD

⇒

⇒

Distance/length of DA

⇒

⇒

Now, by LHS

= AB² + BC² + CD² + DA²

Where are angle between vectors

Take RHS

AC² + BD² + 4PQ²

Thus LHS = RHS

Hence proved