British flag theorem for Rectangle

BRITISH FLAG THEOREM

Point P is inside a rectangle ABCD . P is joined with each vertex . Then Prove that 

AP² +BP² = CP² + DP²

  Sol:-

First of all draw perpendicular on each side from point P .

Now in ∆APE :-

AP² = AE² + PE²             (1)

Now in ∆BPE :-

BP² = BE² + PE²             (2)

Subtracting Eqn. 1 &2 :-

AP²- BP² = AE²- BE²            (3)

Now in ∆DPG :-

DP² = DG² + PG²             (4)

Now in ∆CPG :-

CP² = CG² + PG²             (5)

Subtracting Eqn. 4 FROM 5 :-

CP²- DP² = CG²- DG²            (6)

 Now adding eqn. 3 & 6

AP²- BP² + CP²- DP² = AE²- BE² + CG²- DG²           

( Now AE = DG , BE= CG because AB= CD and perpendicular drawn will divide it in same ratio)

 So      AP²- BP² + CP²- DP² =0

AP² +BP² = CP² + DP²  

                                                                                                                                     ^(H.P.)