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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]
A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
The fact that this square is a pandiagonal magic square can be verified by checking that all of its broken diagonals add up to the same constant: 3+12+14+5 = 34 10+1+7+16 = 34 10+13+7+4 = 34. One way to visualize a broken diagonal is to imagine a "ghost image" of the panmagic square adjacent to the original:
Denote the segments that the diagonal intersection P divides diagonal AC into as AP = p 1 and PC = p 2, and similarly P divides diagonal BD into segments BP = q 1 and PD = q 2. Then the quadrilateral is tangential if and only if any one of the following equalities are true: [ 30 ]
The area K of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals p and q: [8] K = p q 2 . {\displaystyle K={\frac {pq}{2}}.} Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal. [ 6 ]
Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. [1] The fraction 99 / 70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.
An arbitrary quadrilateral and its diagonals. Bases of similar triangles are parallel to the blue diagonal. Ditto for the red diagonal. The base pairs form a parallelogram with half the area of the quadrilateral, A q, as the sum of the areas of the four large triangles, A l is 2 A q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A s is a quarter of A ...