QA-1: Systematics for describing QA-points

In this Encyclopedia of Quadri-Figures 2 coordinate systems are used for Quadrangles:

1. QA-CT-Coordinate system, where 3 arbitrary points of the quadrangle form a Component Triangle (CT). This Component Triangle is defined as Reference Triangle with vertice coordinates (1:0:0), (0:1:0), (0:0:1). The 4^th point is defined as (p:q:r).
2. QA-DT-Coordinate system, where the QA-Diagonal Triangle (DT, see QA-Tr1) is defined as the Reference Triangle with vertice coordinates (1:0:0), (0:1:0), (0:0:1). An arbitrary point of the Quadrangle is defined as (p:q:r).

The other 3 points now form the Anticevian triangle of Pi wrt the QA-Diagonal Triangle and have vertices (-p : q a: r), (p : -q : r), (p : q : -r).
Both coordinate systems can be converted in each other (see QA-6 and QA-7).
Every constructed object now can be identified as:
( f(a,b,c, p,q,r) : f(b,c,a, q,r,p) : f(c,a,b, r,p,q) )
where a,b,c represent the side lengths of the CT- or DT-triangle and
where p,q,r represent the barycentric coordinates wrt the CT- or DT-triangle.
In the description of the points on the following pages only the first of the 3 barycentric coordinates will be shown. The other 2 coordinates can be derived by cyclic rotations:

• a > b > c > a > etc.
• p > q > r > p > 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)).

Transformed Quadrangles
In the descriptions of Quadrangle Centers often the technique is used of transforming one Quadrangle into another Quadrangle. This is done by performing a Transformation T on Pi wrt triangle Pj.Pk.Pl (for all combinations of (i,j,k,l) (1,2,3,4)). This produces a new quadrangle P1’.P2’.P3’.P4’.
Consequently the same transformation T can be performed on quadrangle P1’.P2’.P3’.P4’ producing another quadrangle P1’’.P2’’.P3’’.P4’’. This quadrangle is called the 2^nd generation T-Quadrangle.
It is special that often the Reference Quadrangle and the 2^nd generation T-Quadrangle are homothetic. Consequently, this produces a “Center of Homothecy”, in this paper also named “Homothetic Center”.
Another technique of quadrangle transformation is by determining Triangle Centers (see [12]) Xi for 3 points Pj, Pk, Pl (for all combinations (i,j,k,l) (1,2,3,4)). This produces the (1^st) X-Quadrangle.
The same process can be performed on the (1^st) X-Quadrangle producing the 2^nd X-Quadrangle. This quadrangle is named the 2^nd generation X-Quadrangle.
Again often the Reference Quadrangle and the 2^nd generation X-Quadrangle are homothetic. Again, this produces a “Center of Homothecy”, in this paper also named “Homothetic Center”.

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