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

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SUMMARY

The discussion focuses 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 \( (4 + 2\sqrt{2})S \). The proof involves geometric inequalities and properties of convex shapes. Participants emphasize the importance of understanding the relationship between area and perimeter in quadrilaterals, particularly in the context of optimizing geometric configurations.

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  • Familiarity with geometric inequalities, specifically the triangle inequality
  • Knowledge of area calculations for quadrilaterals
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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:
$Use\,\,area\,\,of \,\,a\,\,triangle=\dfrac {bc\,sin \,A}{2}=---,and\,\, AP\geq GP$
 

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