Maximum and Minimum Properties
Among all quadrilaterals with a given perimeter, the one with the largest area is the square. This is called the isoperimetric theorem for quadrilaterals. It is a direct consequence of the area inequality
where K is the area of a convex quadrilateral with perimeter L. Equality holds if and only if the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter.
The quadrilateral with given side lengths that has the maximum area is the cyclic quadrilateral.
Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area. This is a direct consequence of the fact that the area of a convex quadrilateral satisfies
where θ is the angle between the diagonals p and q. Equality holds if and only if θ = 90°.
If P is an interior point in a convex quadrilateral ABCD, then
From this inequality it follows that the point inside a quadrilateral that minimizes the sum of distances to the vertices is the intersection of the diagonals. Hence that point is the Fermat point of a convex quadrilateral.
Read more about this topic: Quadrilateral
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