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A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following: [6] [7] a parallelogram in which a diagonal bisects an interior angle; a parallelogram in which at least two consecutive sides are equal in length; a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
A quadrilateral is equidiagonal if and only if [5]: p.19, [4]: Cor.4 =. This is a direct consequence of the fact that the area of a convex quadrilateral is twice the area of its Varignon parallelogram and that the diagonals in this parallelogram are the bimedians of the quadrilateral.
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram. The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the ...
By using the area formula of the general rhombus in terms of its diagonal lengths and : The area of the golden rhombus in terms of its diagonal length d {\displaystyle d} is: [ 6 ] A = ( φ d ) ⋅ d 2 = φ 2 d 2 = 1 + 5 4 d 2 ≈ 0.80902 d 2 . {\displaystyle A={{(\varphi d)\cdot d} \over 2}={{\varphi } \over 2}~d^{2}={{1+{\sqrt {5}}} \over 4 ...
For a cyclic orthodiagonal quadrilateral (one that can be inscribed in a circle), suppose the intersection of the diagonals divides one diagonal into segments of lengths p 1 and p 2 and divides the other diagonal into segments of lengths q 1 and q 2. Then [10] (the first equality is Proposition 11 in Archimedes' Book of Lemmas)
The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio, φ, so that the acute angles on each face measure 2 arctan( 1 / φ ) = arctan(2), or approximately 63.43°. A rhombus so obtained is called a golden rhombus.
“It only takes a few norovirus particles to make you and other people sick.” The virus causes acute gastroenteritis — an inflammation of the stomach or intestines — which causes the ...
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 ...