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A square has a larger area than any other quadrilateral with the same perimeter. [7] A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon). The square is in two families of polytopes in two dimensions: hypercube and the cross-polytope. The Schläfli symbol for the square ...
The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. [12] They intersect at the "vertex centroid" of the quadrilateral (see § Remarkable points and lines in a convex quadrilateral below).
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.
The two line segments connecting opposite points of tangency have equal length. The tangent lengths, distances from a point of tangency to an adjacent vertex of the quadrilateral, are equal at two opposite vertices of the quadrilateral. (At each vertex, there are two adjacent points of tangency, but they are the same distance as each other from ...
The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles. [16]: p. 26 One bimedian divides the quadrilateral into two quadrilaterals of equal areas. [16]: p. 26 Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.
The family of lines formed by the sides of a regular polygon together with its axes of symmetry, and; The sides and axes of symmetry of an even regular polygon, together with the line at infinity. Additionally there are many other examples of sporadic simplicial arrangements that do not fit into any known infinite family. [22]
Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent. An object is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes, and have the same size.