QL-3QG1: QL-Component Quadrigons

 
In a Quadrilateral the basic lines L1, L2, L3, L4 are random and have no order.
In a Quadrigon the lines L1, L2, L3, L4 have a fixed order.
Four lines can be ordened and “cycled” in 6 ways (starting with L1):
  1. L1-L2-L3-L4
  2. L1-L2-L4-L3
  3. L1-L3-L2-L4
  4. L1-L3-L4-L2 = reversed sequence of L1-L2-L4-L3
  5. L1-L4-L2-L3 = reversed sequence of L1-L3-L2-L4
  6. L1-L4-L3-L2 = reversed sequence of L1-L2-L3-L4
When we take into account that some sequences are reversed sequences, only 3 types of Quadrigon-orders remain: L1-L2-L3-L4, L1-L2-L4-L3 and L1-L3-L2-L4.
This means that a system of 4 random lines - also named (Complete) Quadrangle - consists of 3 Quadrigons: L1-L2-L3-L4, L1-L2-L4-L3 and L1-L3-L2-L4.
Note that a Quadrigon can be defined by a set of two opposite lines.
The distinction of 3 Component Quadrigons is important because in some constructions there is an order in reference lines and in other constructions not.
When there is no order in reference lines then the barycentric coordinates of derived centers and other items will be symmetric. When there is a given order in reference lines then the barycentric coordinates of derived centers and other items will not be symmetric.
Example: In a Quadrigon we have 2 sets of opposite lines. The line connecting the intersection points of these 2 sets of opposite lines is called the 3rd diagonal of a Quadrigon (QG-L1). Since a Quadrilateral consists of 3 Quadrigons there are three 3rd diagonals forming the QL-Diagonal Triangle (QL-Tr1).
 
QL-3QG1-Component Quadrigons-01
 

 
 

 

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