Prove Quadrilateral ABCD Perimeter $\geq (4+2\sqrt 2)S$

[Albert1]

A convex quadrilateral ABCD with area $S^2$ , prove the sum of its perimeter and two diagonal lines $\geq (4+2\sqrt 2)S$

 

[Albert1]

A convex quadrilateral ABCD with area $S^2$ , prove the sum of its perimeter and two diagonal lines $\geq (4+2\sqrt 2)S$

hint:

Spoiler

$Use\,\,area\,\,of \,\,a\,\,triangle=\dfrac {bc\,sin \,A}{2}=---,and\,\, AP\geq GP$