Quadrilaterals

A systematic exploration.

## Notation

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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

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### Excluded properties

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* Non-euclidian
* Non-planar

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

### Edge-case properties

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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

### Vertex properties

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### Length properties

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* 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

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* 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

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* 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

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

### Compound properties

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