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The rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of equal sides. The parallel sides differ in length by units where: = (+) It will be easier in this case to revert to the standard statement of Ptolemy's theorem:
A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol {2}. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated digon, t{2} is a square, {4}.
Similarity tests look at whether the ratios of the lengths of each pair of corresponding sides are equal, though again this is not sufficient. In either case equality of corresponding angles is also necessary; equality (or proportionality) of corresponding sides combined with equality of corresponding angles is necessary and sufficient for ...
The only equable rectangles with integer sides are the 4 × 4 square and the 3 × 6 rectangle. [4] An integer rectangle is a special type of polyomino, and more generally there exist polyominoes with equal area and perimeter for any even integer area greater than or equal to 16. For smaller areas, the perimeter of a polyomino must exceed its area.
While the circle has a relatively low maximum packing density, it does not have the lowest possible, even among centrally-symmetric convex shapes: the smoothed octagon has a packing density of about 0.902414, the smallest known for centrally-symmetric convex shapes and conjectured to be the smallest possible. [3]
A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides. The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a ...
An antiparallelogram is a special case of a crossed quadrilateral, with two pairs of equal-length edges. [3] In general, crossed quadrilaterals can have unequal edges. [ 3 ] Special forms of the antiparallelogram are the crossed rectangles and crossed squares, obtained by replacing two opposite sides of a rectangle or square by the two diagonals.
The root-3 rectangle is also called sixton, [6] and its short and longer sides are proportionally equivalent to the side and diameter of a hexagon. [ 7 ] Since 2 is the square root of 4, the root-4 rectangle has a proportion 1:2, which means that it is equivalent to two squares side-by-side.