PRINCIPIA · DEFINITION

Rhombus

A parallelogram whose four sides are all equal; from this its two diagonals are perpendicular bisectors of each other and each bisects a pair of opposite angles.

Dependencies: Parallelogram, Distance. A derived definition: add to a parallelogram the condition that one pair of adjacent sides has equal length.

Definition

Let ABCDABCD be a parallelogram.

Rhombus ABCD has an equal-length mark on each of its four sides.

If one pair of adjacent sides has equal length, then the parallelogram is called a rhombus.

Opposite sides of a parallelogram are already equal. Making one adjacent pair equal therefore makes all four sides equal. A rhombus emphasizes equal sides just as a rectangle emphasizes right angles.

Remarks

  • One equal adjacent pair is enough. Opposite sides are already equal in a parallelogram, so the definition forces all four sides to be equal.
  • Perpendicular bisecting diagonals. The diagonals of a rhombus perpendicularly bisect each other, and each bisects a pair of opposite angles (Rhombus: diagonals ⊥-bisect).
  • A square is a special case. Add one right angle, making the rhombus also a rectangle, and the result is a square.
  • Incircle. The intersection of the diagonals is equidistant from all four sides, so a rhombus has an incircle centered there.