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Concept
Rhombus
Summary
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.
Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square.
The word "rhombus" comes from rhombos, meaning something that spins, which derives from the verb ῥέμβω, romanized: , meaning "to turn round and round." The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for a bicone, two right circular cones sharing a common base.
The surface we refer to as rhombus today is a cross section of the bicone on a plane through the apexes of the two cones.
A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:
a parallelogram in which a diagonal bisects an interior angle
a parallelogram in which at least two consecutive sides are equal in length
a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
a quadrilateral with four sides of equal length (by definition)
a quadrilateral in which the diagonals are perpendicular and bisect each other
a quadrilateral in which each diagonal bisects two opposite interior angles
a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent
a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point
Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides.
Official source
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