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  2. Euler's quadrilateral theorem - Wikipedia

    en.wikipedia.org/wiki/Euler's_quadrilateral_theorem

    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 .

  3. Brahmagupta's formula - Wikipedia

    en.wikipedia.org/wiki/Brahmagupta's_formula

    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.

  4. Cyclic quadrilateral - Wikipedia

    en.wikipedia.org/wiki/Cyclic_quadrilateral

    A set of sides that can form a cyclic quadrilateral can be arranged in any of three distinct sequences each of which can form a cyclic quadrilateral of the same area in the same circumcircle (the areas being the same according to Brahmagupta's area formula). Any two of these cyclic quadrilaterals have one diagonal length in common. [17]: p. 84

  5. Quadrilateral - Wikipedia

    en.wikipedia.org/wiki/Quadrilateral

    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.

  6. Tangential quadrilateral - Wikipedia

    en.wikipedia.org/wiki/Tangential_quadrilateral

    For a tangential quadrilateral with given sides, the inradius is maximum when the quadrilateral is also cyclic (and hence a bicentric quadrilateral). In terms of the tangent lengths, the incircle has radius [8]: Lemma2 [14] = + + + + + +.

  7. Isosceles trapezoid - Wikipedia

    en.wikipedia.org/wiki/Isosceles_trapezoid

    Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. [5] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms ...

  8. Bretschneider's formula - Wikipedia

    en.wikipedia.org/wiki/Bretschneider's_formula

    Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.. The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals e and f to give [2] [3]

  9. Midsquare quadrilateral - Wikipedia

    en.wikipedia.org/wiki/Midsquare_quadrilateral

    Midsquare quadrilaterals whose sides are not longer than the diagonals have the maximum area for their diameter among all quadrilaterals, solving the = case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others.