He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral.

Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.

If rectangles are included in the class of trapezoids then one may concisely define an isosceles trapezoid as " a cyclic quadrilateral with equal diagonals " or as " a cyclic quadrilateral with a pair of parallel sides.

A convex quadrilateral is also a trapezoid if and only if the diagonals cut the quadrilateral into four triangles of which one opposite pair are similar.

If the trapezoid is divided into four triangles by its diagonals AC and BD ( as shown on the right ), intersecting at O, then the area of is equal to that of, and the product of the areas of and is equal to that of and.

For the sum of the diagonals we have the inequality

Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality.

The exact reason for this split has been a point of contention among art historians ; usually the divergent ideas about the directions of the lines in the paintings have been named as the primary reason: Mondrian never accepted diagonals, whereas Doesburg insisted on the diagonal's dynamic aspects, and indeed featured it in his art.

Any four exterior vertices determine a cyclic quadrilateral, and all cyclic quadrilaterals are convex quadrilaterals, so each set of four exterior vertices have exactly one point of intersection formed by their diagonals ( chords ).

For instance, a cubic crystal may have low-energy planes on the faces of the cube or on the diagonals.

* The diagonals have the same length.

As pictured, the diagonals AC and BD have the same length () and divide each other into segments of the same length ( and ).

In mathematics, an Euler brick, named after Leonhard Euler, is a cuboid whose edges and face diagonals all have integer lengths.

These variations have the leg pairs across the diagonals.

In some computer-generated puzzles, if the person solving the puzzle sees one word, all they have to do to find more is to look in adjacent rows, columns, or diagonals.

While, it is possible that the originators of the design may not have been aware of the particular proportions they were generating as they worked, it's more likely that the methods of construction using diagonals and curves would have taught them something.

* 2 nonominoes have four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4.

Most modern joysticks have an 8-way configuration, allowing for movement in the cardinal directions and the diagonals.

In recreational mathematics, a magic square of order n is an arrangement of n < sup > 2 </ sup > numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.

Another necessary and sufficient condition is that the diagonals cut each other in mutually the same ratio ( this ratio is the same as that between the lengths of the parallel sides ).

In chess, the distance between squares on the chessboard for rooks is measured in Manhattan distance ; kings and queens use Chebyshev distance, and bishops use the Manhattan distance ( between squares of the same color ) on the chessboard rotated 45 degrees, i. e., with its diagonals as coordinate axes.

Moreover, the diagonals divide each other in the same proportions.

Particles along the same horizontal line share the same Strangeness ( particle physics ) | strangeness, s, while those on the same diagonals share the same electric charge | charge, q.

Since the diagonals AC and BD divide each other into segments of equal length, the diagonals bisect each other.

* The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.

The diagonals of a cube with side length 1.

where x is the length of the line joining the midpoints of the diagonals.

In the middle of the flag is a white circle whose diameter is equal to one-third of the length of the rectangle and whose center is located at the intersection of the diagonals of the rectangle.

The diagonals are also of equal length.

* The diagonals of a square are ( about 1. 414 ) times the length of a side of the square.

The ancient geometers are not done yet, for if the fifth vertex of the pentagon is marked as E and FE and BF are joined ( with FE = BF = z ), then cyclic quadrilateral EFBA will be formed with diagonals length d ( diameter ) and b. Applying the ' Almagest ' theorem yet again:

When applied repeatedly, Ptolemy's theorem allows one to compute the lengths of all diagonals for polygons inscribed in a circle with vertices P < sub > 1 </ sub >, ..., P < sub > n </ sub >, if the sides are given together with all the length values of the " next to sides " chords connecting two vertices P < sub > i </ sub > and P < sub > i + 2 </ sub >

The difference in length of both diagonals and the illumination gradient, are both classic indications of an out-of-level sample.

* < span style =" color :# 00f ;"> Blue-Rebound Laser </ span >-fires three thin beams ; one horizontal, two at 45 ° diagonals.

To this end, Beck devised a simplified map, consisting of stations, straight line segments connecting them, and the River Thames ; lines ran only vertically, horizontally, or on 45 degree diagonals.

Scott's residential buildings are few ; one of the best known is the Cropthorne Court mansion block in Maida Vale, where the frontage juts out in diagonals in order to eliminate the need for lightwells.

# The two diagonals of a rhombus are perpendicular ; that is, a rhombus is an orthodiagonal quadrilateral.

A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel ; adjacent angles are supplementary ; the two diagonals bisect one another ; any line through the midpoint bisects the area ; and the sum of the squares of the sides equals the sum of the squares of the diagonals ( the parallelogram law ).

In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles ; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.

The two segments of the two diagonals are two sides in a triangle ; the base the quadrilateral is the base of the triangle.

In mathematics, a magic hypercube is the k-dimensional generalization of magic squares, magic cubes and magic tesseracts ; that is, a number of integers arranged in an n × n × n × ... × n pattern such that the sum of the numbers on each pillar ( along any axis ) as well as the main space diagonals is equal to a single number, the so-called magic constant of the hypercube, denoted M < sub > k </ sub >( n ).

The 16 positions lying at the 4 space diagonals ( 8 corners and 8 internal positions ) are equivalent and each involved in 7 winning lines ; the other 48 positions ( 24 face positions and 24 edge positions ) are also equivalent, each being involved in four winning lines.

* The diagonals divide each other into segments with lengths that are pairwise equal ; in terms of the picture below,, ( and if one wishes to exclude rectangles ).

The maximum possible number of triangles in a simple arrangement is known to be upper bounded by n ( n − 1 )/ 3 and lower bounded by n ( n − 3 )/ 3 ; the lower bound is achieved by certain subsets of the diagonals of a regular 2n-gon.

Jabodetabek on left in blue and magenta ; Greater Bandung on right, Jakarta and 4 kotas in blue, 3 suburban regencies in magenta, green diagonals mark sprawl areas outside Jabodetabek: Serang and Karawang Regencies

also knew of this sequence of approximations ; they called the denominators and numerators of this sequence side and diameter numbers and the numerators were also known as rational diagonals or rational diameters.

i. e. Broken diagonals are 1-D in a 2_D square ; broken oblique squares are 2-D in a 3-D cube.

The main dome rests on four semi-domes ; not on the axes but in the diagonals of the building.