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

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The discussion centers on proving that for a convex quadrilateral ABCD with area S^2, the sum of its perimeter and the lengths of its two diagonals is at least (4 + 2√2)S. Participants explore geometric properties and inequalities related to quadrilaterals, emphasizing the relationship between area and perimeter. Various approaches to the proof are suggested, including the use of known inequalities and geometric constructions. The hint provided encourages considering specific configurations of the quadrilateral to facilitate the proof. Ultimately, the goal is to establish a rigorous mathematical foundation for the inequality in question.
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A convex quadrilateral ABCD with area $S^2$ , prove the sum of its perimeter and two diagonal lines $\geq (4+2\sqrt 2)S$
 
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Albert said:
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:
$Use\,\,area\,\,of \,\,a\,\,triangle=\dfrac {bc\,sin \,A}{2}=---,and\,\, AP\geq GP$
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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