In a quadrilateral PQRS the diagonals PR and QS intersect at O. Prove that

i)PQ + QR + RS + +SP PR +QS

ii)PQ + QR + RS + SP

Proof:
....In tri.POQ, PO+OQ>PQ------(1)
....In tri.QOR, QO+OR>QR------(2)
....In tri.ROS, RO+OS>RS-------(3)
....In tri.SOR, SO+OP>SP-------(4)
(The sum of any 2 sides> the 3rd side in length)
1. PQ+PS>QS
....QR+RS>QS
(The sum of any 2 sides> the 3rd side in length)
adding both equations,

We get ,

= PQ+PS+QR+RS>2QS-------(5)

Now,
....PQ+QR>PR
....PS+SR>PR
(The sum of any 2 sides>the 3rd side in length)
=>PQ+PS+QR+RS>2PR-------(6)
adding (5) and (6):
2(PQ+PS+QR+RS)>2(QS+PR)=>
PQ+PS+QR+RS>QS+PR  (proved)

2. (1)+(2)+(3)+(4):
2(PO+QO+OR+SO)>PQ+QR+RS+SP=>
2(PR+QS)>PQ+QR+RS+SP (HENCE PROVED)