QA-3QG1: QA-Component Quadrigons
In a Quadrangle the vertices P1, P2, P3, P4 are random and have no order.
In a Quadrigon the vertices P1, P2, P3, P4 have a fixed order.
Four points can be ordened and “cycled” in 6 ways (starting with P1):
- P1-P3-P4-P2 = reversed sequence of P1-P2-P4-P3
- P1-P4-P2-P3 = reversed sequence of P1-P3-P2-P4
- P1-P4-P3-P2 = reversed sequence of P1-P2-P3-P4
When we take into account that some sequences are reversed sequences, only 3 types of Quadrigon-orders remain: P1-P2-P3-P4, P1-P2-P4-P3 and P1-P3-P2-P4.
This means that a system of 4 random points - also named (Complete) Quadrangle - consists of 3 Quadrigons: P1-P2-P3-P4, P1-P2-P4-P3 and P1-P3-P2-P4.
Note that a Quadrigon can be defined by a set of two opposite points.
The distinction of 3 Component Quadrigons is important because in some constructions there is an order in reference points and in other constructions not.
When there is no order in reference points then the barycentric coordinates of derived centers will be symmetric. When there is a given order in reference points then the barycentric coordinates of derived centers will not be symmetric.
A simple example is that a Quadrigon has 2 diagonals because there are 2 sets of opposite points. However a Quadrangle has no opposite points so there are no conventional diagonals. Meanwhile a Quadrangle has three Diagonal Crosspoints (one per component Quadrigon) forming the QA-Diagonal Triangle (QA-Tr1).