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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. [43] Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area.
Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem .
In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral.
Brahmagupta's theorem states that for a cyclic quadrilateral that is also orthodiagonal, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side. [23] If a cyclic quadrilateral is also orthodiagonal, the distance from the circumcenter to any side equals half the length of the opposite side. [23]
Brahmagupta's theorem states that for a cyclic orthodiagonal quadrilateral, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side. [3] If an orthodiagonal quadrilateral is also cyclic, the distance from the circumcenter (the center of the circumscribed circle) to any side equals half the ...
Fuss' theorem for the relation among the same three variables in bicentric quadrilaterals; Poncelet's closure theorem, showing that there is an infinity of triangles with the same two circles (and therefore the same R, r, and d) Egan conjecture, generalization to higher dimensions; List of triangle inequalities
For given side lengths, the area is maximum when the quadrilateral is also cyclic and hence a bicentric quadrilateral. Then = since opposite angles are supplementary angles. This can be proved in another way using calculus. [12]
A bicentric quadrilateral has a greater inradius than does any other tangential quadrilateral having the same sequence of side lengths. [ 20 ] : pp.392–393 Inequalities