QL-Tr-1: QL-Triple Triangles


Triple Triangles / Triple Lines
The four lines L1, L2, L3, 4 in a Quadrilateral can be placed in 3 different cyclic sequences.
These cyclic sequences are:
• L1 – L2 – L3 – L4,
• L1 – L2 – L4 – L3,
• L1 – L3 – L2 – L4.
Each of these 3 cyclic 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.
See also explanantion at QG-P1 and QA-Tr-1.

Special properties related to Triple Triangles
Triple Triangles are because of their construction special triangles.
They often have an intrinsic structure which lead to special properties.
There are many examples of incidences with QL-points.
Eckart Schmidt examined following QL-Triple Triangles coming from QG-points. See Ref-66, QPG-message #422.

Flat QL-Triple Triangles
The vertices of the Triple Triangles of QG-P2/QG-P10/QG-P15/QG-P16/QG-P18 are flat/collinear.
... for QG-P2 on QL-L1,
... for QG-P10 on QL-L6,
... for QG-P12 on QL-L1,
... for QG-P15 on QL-L9,
... for QG-P16 on QL-L11,
... for QG-P18 on CSC(QL-Ci2) .

Perspective QL-Triple Triangles
QL-triple triangles perspective QL-Tr1:
... for QG-P3 perspector QL-P8,
... ... perspectrix line at infinity,
... for QG-P5 perspector QL-P16,
... ... perspectrix QL-L1,
... for QG-P6 perspector infinity point of QL-L2,
... ... perspectrix QL-L6,
... for QG-P9 perspector QL-P16,
... ... perspectrix parallel QL-P17.23
... ... ... through intersection of QL-L1 and bisector of QL-P17.23,
... for QG-P10 perspector infinity point of QL-L2,
... ... perspectrix QL-L6,
... for QG-P12 perspector QL-P13,
... ... perspectrix QL-L1,
... for QG-P13 perspector QL-P13,
... ... perspectrix QL-L1,
... for QG-P17 perspector QL-P10,
... ... perspectrix radical axis of QL-Ci1 and polarcircle of Ql-Tr1,
... for QG-P19 perspector ..., perspectrix ..., see also Ref-34, QFG message #3176.