QL-1: Systematics for describing QL-Points

The method for describing points, lines, and other curves related to quadrilaterals involves referencing lines rather than points (as is done when describing a quadrangle). The major advantage of this approach is that all resulting algebraic expressions are symmetric.

This system is based on three lines of the quadrilateral, represented with barycentric homogeneous coefficients:

• L₁: (1:0:0)
• L₂: (0:1:0)
• L₃: (0:0:1)

The fourth line is assigned coefficients (l:m:n).

Note: The notation (l:m:n) is used for lines, whereas (p:q:r) is reserved exclusively for points.

Coordinate Systems Used

In this Encyclopedia of Quadri-Figures, two coordinate systems are used for quadrilaterals:

1. QL-CT Coordinate System
   Three arbitrary lines of the quadrilateral form a Component Triangle (CT). This triangle is defined as the reference triangle with line coefficients:
   • (1:0:0)
   • (0:1:0)
   • (0:0:1)

   The fourth line is then defined as (l:m:n).

2. QL-DT Coordinate System
   The QL-Diagonal Triangle (DT, see QL-Tr1) is defined as the reference triangle with the same line coefficients:
   • (1:0:0)
   • (0:1:0)
   • (0:0:1)

   An arbitrary line of the quadrilateral is defined as (l:m:n). The remaining three lines now form the lines of the Anticevian Triangle with respect to the QL-Diagonal Triangle. Each line Lᵢ acts as a perspectrix and has coefficients:

   • (-l : m : n)
   • (l : -m : n)
   • (l : m : -n)

Both coordinate systems are interconvertible (see QL-6 and QL-7).

Every constructed object can now be identified as:

( f(a,b,c, l,m,n) : f(b,c,a, m,n,l) : f(c,a,b, n,l,m) )

where a,b,c represent the side lengths of the triangle with vertices P1 (1:0:0), P2 (0:1:0) and P3 (0:0:1).

In the description of the points on the following pages only the first of the 3 barycentric coefficients will be shown. The other 2 coordinates can be derived by cyclic rotations:

• a > b > c > a > etc.
• l > m > n > l > etc.

Further the Conway notation has been used in algebraic expressions:

• S_A = (-a² + b² + c²) / 2
• S_B = (+a² – b² + c²) / 2
• S_C = (+a² + b² – c²) / 2
• S_ω = (+a² + b² + c²) / 2
• S = √( S_A S_B + S_B S_C + S_C S_A) = 2 Δ

where Δ = area triangle ABC = ¼ √((a+b+c) (-a+b+c) (a-b+c) (a+b-c))

Triple Triangles / Triple Lines

The four lines L1, L2, L3, L4 in a Quadrangle can be placed in 3 different cyclic sequences.

These sequences are:

• L1 – L2 – L3 – L4,
• L1 – L2 – L4 – L3,
• L1 – L3 – L2 – L4.

Each of these 3 sequences represent a Quadrigon.

Just like a Quadrilateral has 4 Component Triangles, it also has 3 Component Quadrigons.

In these Component Quadrigons points can be constructed belonging to the domain of a Quadrigon or a Quadrangle. This gives us per Quadrilateral 3 versions of a Quadrigon Point / Quadrangle Point.

These 3 points form a Triple. The Triangle formed by this Triple is called the “QG-Px-Triple Triangle” or “QA-Px-Triple Triangle” in a Quadrilateral.

When the 3 points are collinear the Triple forms a Line called the “QG-Px-Triple Line” or “QA-Px-Triple Line” in a Quadrilateral.

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