QA-1: Systematics for describing QA-points
In this Encyclopedia of Quadri-Figures 2 coordinate systems are used for Quadrangles:
-
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 barycentric vertice coordinates (1:0:0), (0:1:0), (0:0:1). The 4th point is defined as (p:q:r).
-
QA-DT-Coordinate system, where the QA-Diagonal Triangle (DT, see QA-Tr1) is defined as the Reference Triangle with barycentric 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 : r), (p : -q : r), (p : q : -r).
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:
- 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)).
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 permutations 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 2nd generation T-Quadrangle.
It is special that often the Reference Quadrangle and the 2nd generation T-Quadrangle are homothetic. Consequently, this produces a “Center of Homothecy”, in EQF also named “Homothetic Center”.
Another technique of quadrangle transformation is by determining Triangle Centers (see Ref-12) Xi for 3 points Pj, Pk, Pl (for all permutations (i,j,k,l)∈ (1,2,3,4)). This produces the (1st) X-Quadrangle.
The same process can be performed on the (1st) X-Quadrangle producing the 2nd X-Quadrangle. This quadrangle is named the 2nd generation X-Quadrangle.
Again often the Reference Quadrangle and the 2nd generation X-Quadrangle are homothetic. Again, this produces a “Center of Homothecy”, in EQF also named “Homothetic Center”.
Triple Triangles / Triple Lines
The four points P1, P2, P3, P4 in a Quadrangle can be placed in 3 different cyclic sequences.
These sequences are:
- P1 – P2 – P3 – P4,
- P1 – P2 – P4 - P4,
- P1 – P3 – P2 – P4.
Each of these 3 sequences represent a Quadrigon.
Just like a Quadrangle 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 Quadrilateral. This gives us per Quadrangle 3 versions of a Quadrigon Point / Quadrilateral Point.
These 3 points form a Triple. The Triangle formed by this Triple is called the “QG-Px-Triple Triangle” or “QL-Px-Triple Triangle” in a Quadrangle.
When the 3 points are collinear the Triple forms a Line called the “QG-Px-Triple Line” or “QL-Px-Triple Line” in a Quadrangle.