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A square is a limiting case of both a kite and a rhombus. Orthodiagonal equidiagonal quadrilaterals in which the diagonals are at least as long as all of the quadrilateral's sides have the maximum area for their diameter among all quadrilaterals, solving the n = 4 case of the biggest little polygon problem. The square is one such quadrilateral ...
Another way of saying the same thing is that the center points of the four squares form the vertices of an equidiagonal orthodiagonal quadrilateral. The theorem is named after Belgian mathematician Henricus Hubertus (Henri) Van Aubel (1830–1906), who published it in 1878. [1]
In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to = since θ is 90°. The area can be also expressed in terms of bimedians as [16] = , where the lengths of the bimedians are m and n and the angle between them is φ.
The condition that p = q means that the quadrilateral is an equidiagonal quadrilateral (its diagonals have equal length), and the condition that sin(θ) = 1 means that it is an orthodiagonal quadrilateral (its diagonals cross at right angles). The quadrilaterals of this type include the square with unit-length diagonals, which has area 1/2 ...
Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential.
The square is one such quadrilateral, but there are infinitely many others. Equidiagonal, orthodiagonal quadrilaterals have been referred to as midsquare quadrilaterals [ 4 ] : p. 137 because they are the only ones for which the Varignon parallelogram (with vertices at the midpoints of the quadrilateral's sides) is a square.
The square has Dih 4 symmetry, order 8. There are 2 dihedral subgroups: Dih 2, Dih 1, and 3 cyclic subgroups: Z 4, Z 2, and Z 1. A square is a special case of many lower symmetry quadrilaterals: A rectangle with two adjacent equal sides; A quadrilateral with four equal sides and four right angles; A parallelogram with one right angle and two ...
In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. [1] It is named after the Indian mathematician Brahmagupta (598-668). [2]