Quadrilaterals

A systematic exploration.

Notation

This exploration will use a special postfix notation to describe quadrilateral properties. It may evolve over time. If successful, it may become useful for more polytopes.

The Vertices are labelled A, B, C and D, starting in a clockwise manner.

The sides are labelled as a, b, c, d, starting clockwise and beginning from A. The diagonals are labeled e (A to C) and f (B to D)

The angles are labelled α, β, γ, δ starting internally at A and following on the same side. In complex quadrilaterals, that means 'external-looking' angles are measured after the intersection of sides.

The canonical form of properties is their shortest and lexicographically earliest form. Vertices before sides before angles.

Properties

Excluded properties

• Non-euclidian
• Non-planar

Going forth, quadrilaterals will be assumed to always be on an euclidean plane.

Edge-case properties

These properties may disqualify a quadrilaterals from being considered true quadrilaterals and instead being considered edge-cases or 'degenerate'.

• Coinciding vertices: two or more vertices in the same place
  • Neighboring vertices coinciding: one side of zero length, angles at coinciding vertices are undefined
  • Two pairs of neighboring vertices coinciding: two sides of zero length; all angles undefined; zero area
  • Opposing vertices coinciding: zero area, two zero angles
  • Three vertices coinciding: two sides of zero length; one zero angle; three angles undefined; zero area
  • All vertices coinciding: all sides have zero length; all angles undefined; zero area
• Zero-angle
• Straight angle
• Zero length side: same as two neighboring vertices coinciding
• Zero length diagonal: same as opposite vertices coinciding

Length properties

• Equal length sides
  • Two adjacent sides are equal length
  • Two opposing sides are equal length Watt quadrilateral (Q98098837)
  • Three sides are equal length
  • Two separate pairs of adjacent sides are equal length: deltoid (Q78329344)
    • If convex: kite kite (Q107061)
    • if concave: dart dart (Q18043403)
  • Two opposing pairs of sides are equal length
    • if simple: parallelogram (Q45867))
    • of complex: anti-parallelogram antiparallelogram (Q581094)
• Equal length diagonals equidiagonal quadrilateral (Q5384444)
• Bisected diagonals (only possible on simple quadrilaterals)
  • One diagonal bisects the other bisect-diagonal quadrilateral (Q130349161)
  • Both diagonals bisect another: parallelogram (Q45867)

Angle properties

• Equal (main) angles
  • Two neighboring angles are equal
  • Two opposing angles are equal
  • Three equal angles
  • Two pairs of distinct neighboring angles are equal: isosceles trapezoid (Q1194115)
  • Two opposite angles are equal: (Q45867))
  • All angles equal
    • Rectangle rectangle (Q209)
    • Crossed-over rectangles have the same internal angles, but according to convention we consider the external angles after the crossing.
• Right (main) angles
  • One right angle
  • Two neighboring right angles: right trapezoid (Q12218380)
  • Two opposing right angles: Hjelmslev quadrilateral (Q98098744)
  • Three right angles, fourth right angle implied: rectangle
• Right angle between diagonals orthodiagonal quadrilateral (Q3531598)
• Parallels
  • Opposing sides parallel, opposite direction: trapezoid (Q46303)
  • Two opposing sides parallel: parallelogram (Q45867)
  • Opposing sides parallel, same direction: cross-legged trapezoid
  • Parallel diagonals: trapezoid, parallel lines crossed over
• Conjugate angles
  • two pairs of neighboring angles are conjugate: cross-legged trapezoid
  • both opposite angles are conjugate: anti-parallelogram

Symmetries

• One reflection along axis not along diagonal
  • When simple: isosceles trapezoid
• Two reflection along axes that aren't along vertices
  • implies two-fold rotational symmetry
  • if convex: rectangle
• Reflection along one diagonal: deltoid
  • When convex: (true) kite
  • When concave: dart
  • Edge case: straight angled 'dartkite'
• 2 Reflections along diagonal: rhombus
• 1 Reflection along diagonal and one along non-diagonal: square; two more reflection axes and 4-fold rotational symmetry is implied
• 2-fold rotational symmetry
  • When simple: parallelogram
• 4-fold rotational symmetry: square; reflection along 4 axes implied

Circles

• Tangential: all sides are tangential to one circle
• Cyclic: all vertices are on one circle