Search results
Results from the WOW.Com Content Network
This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces. In spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.
The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length , while AC (shown in red) is a face diagonal and has length . In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.
One edge, two face diagonals and the space diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16. Two edges must have length divisible by 3 and at least one of those edges must have length divisible by 9. One edge must have length divisible by 5.
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
If an horizontal line is drawn through the intersection point of the diagonal and the internal edge of the square, the original golden rectangle and the two scaled copies along the diagonal have linear sizes in the ratios ::, the square and rectangle opposite the diagonal both have areas equal to . [10]
where = + is the length of the rectangle's diagonal. If the two points are instead chosen to be on different sides of the square, the average distance is given by [ 3 ] [ 4 ] ( 2 + 2 + 5 ln ( 1 + 2 ) 9 ) s ≈ 0.869009 … s . {\displaystyle \left({\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{9}}\right)s\approx 0.869009\ldots s.}
The inscribed square problem, also known as the square peg problem or the Toeplitz' conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some square? This is true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. [1]
If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well: 2 a 2 + 2 b 2 = 2 e 2 {\displaystyle 2a^{2}+2b^{2}=2e^{2}} Dividing by 2 yields the Euler–Pythagoras theorem: