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A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals [4] (therefore only two sides are parallel). It is a special case of an antiparallelogram , and its angles are not right angles and not all equal, though opposite angles are equal.
A root-phi rectangle divides into a pair of Kepler triangles (right triangles with edge lengths in geometric progression). The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles.
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.
It states that in a cyclic quadrilateral , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference ...
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:
For the general quadrilateral (with four sides not necessarily equal) Euler's quadrilateral theorem states + + + = + +, where is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that x = 0 {\displaystyle x=0} for a parallelogram, and so the general formula simplifies to the parallelogram law.
The massive stock market gains of the past two years — the S&P gained roughly 20% in 2023 and is set to gain more than that by the end of 2024 — also pose challenges to US companies. Benchmark ...
A square has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is 2 ≈ 1.414. {\displaystyle {\sqrt {2}}\approx 1.414.} A regular pentagon has five diagonals all of the same length.