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. M is used for for the intersection of the diagonals (in simple quadrilaterals) or the intersection of two sides (in complex quadrilaterals).

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
* Three sides are equal length
* Two separate pairs of adjacent sides are equal length
* If convex: kite
* if concave: dart
* Two opposing pairs of sides are equal length
* if simple: parallelogram
* of complex: anti-parallelogram

### Angle properties

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### 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
* 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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