QL-1: Systematics for describing QL-Points
The way to describe points, lines and other curves related to Quadrilaterals is by relating to lines instead of points (as done when describing a Quadrangle).
The big advantage is that all resulting algebraic expressions will be symmetric.
The system used is based on 3 lines of the quadrilateral with barycentric homogeneous coefficients L1(1:0:0), L2(0:1:0) and L3(0:0:1). The 4th line gets coefficients (l:m:n).
Note that the line notation (l:m:n) is used instead of (p:q:r) which is only used for points.
In this Encyclopedia of Quadri-Figures 2 coordinate systems are used for Quadrilaterals:
- QL-CT-Coordinate system, where 3 arbitrary lines of the quadrilateral form a Component Triangle (CT). This Component Triangle is defined as Reference Triangle with line coefficients (1:0:0), (0:1:0), (0:0:1). The 4th line is defined as (l:m:n).
- QL-DT-Coordinate system, where the QL-Diagonal Triangle (DT, see QL-Tr1) is defined as the Reference Triangle with sideline coefficients (1:0:0), (0:1:0), (0:0:1). An arbitrary line of the Quadrilateral is defined as (l:m:n). The other 3 lines now form the lines of the Anticevian triangle wrt the QL-Diagonal Triangle where Li is the perspectrix and have line coefficients (-l : m : n), (l : -m : n), (l : m : -n).
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:
- SA = (-a2 + b2 + c2) / 2
- SB = (+a2 - b2 + c2) / 2
- SC = (+a2 + b2 - c2) / 2
- Sω = (+a2 + b2 + c2) / 2
- S = √( SA SB + SB SC + SC SA) = 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 Quadrilateral 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.