Answer :

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


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