QA-Tr-1: QA-Triple Triangles

Triple Triangles / Triple Lines
The four points P1, P2, P3, P4 in a Quadrangle can be placed in 3 different cyclic sequences.
These cyclic sequences are:
• P1 – P2 – P3 – P4,
• P1 – P2 – P4 – P3,
• P1 – P3 – P2 – P4.
Each of these 3 cyclic sequences represent a Quadrigon.
Just like a Quadrangle has 4 Component Triangles, it also has 3 Component Quadrigons.
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.
For a further explanation see QG-P1 and QL-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 of Circumcircles of Triple Triangles with QA-points.
There are many examples where the vertices of a Triple Triangle lie on a special conic, cubic or other curve.
There are flat Triple Triangles, where the vertices are collinear.
There are many examples of homothetic sets of Triple Triangles, perspective pairs of Triple Triangles (see QA-Tr-2), Orthologic pairs of Triple Triangles (see QA-Tr-3), Cyclologic pairs of Triple Triangles (see QA-Tr-4), Parallelologic pairs of Triple Triangles (see QA-Tr-5).
Last but not least there are Triple Triangles that are perspective, Orthologic or Cyclologic with all Component Triangles of the Reference Quadrangle. In this case we call these Triple Triangles (as proposed by Seiichi Kirikami, Ref-34, QFG #979, #980) resp. Quadri-Perspective (or Desmic, see Ref-34, QFG #986), Quadri-Orthologic, Quadri-Cyclologic.

Flat QA-Triple Triangles
The vertices of the Triple Triangles of QG-P3/QL-P5/QL-P15/QL-P18 are flat/collinear.

Homothetic sets of QA-Triple Triangles
FAM-QA1 = Family of Homothetic sets of Triple Triangles of QG-P1/QG-P2/QG-P4/QG-P8/QG-P15/QL-P12
FAM-QA2 = Family of Homothetic sets of Triple Triangles of QG-P3/QL-P15
FAM-QA3 = Family of Homothetic sets of Triple Triangles of QG-P5/QG-P10/QL-P2
FAM-QA4 = Family of Homothetic sets of Triple Triangles of QG-P7/QG-P9/QL-P6
FAM-QA5 = Family of Homothetic sets of Triple Triangles of QG-P12/QG-P14
FAM-QA2 are "Flat" Triangles, their vertices are collinear.
FAM-QA3 and FAM-QA4 are Families of Similar but rotated Triangles.

Definition Desmic Configuration
Three Quadrangles are said to form a Desmic Configuration when the Quadrangles are pairwise perspective in a fourfold way.
When three Quadrangles P1P2P3P4, Q1Q2Q3Q4 and R1R2R3R4 form a Desmic Configuration the perspective organization will be as follows:
Wrt pair of Quadrangles P1P2P3P4 and Q1Q2Q3Q4 (points in given order):
P1P2P3P4 ~ Q1Q2Q3Q4 with perspector R1
P1P2P3P4 ~ Q2Q1Q4Q3 with perspector R2
P1P2P3P4 ~ Q3Q4Q1Q2 with perspector R3
P1P2P3P4 ~ Q4Q3Q2Q1 with perspector R4
Wrt pair of Quadrangles Q1Q2Q3Q4 and R1R2R3R4 (points in given order):
Q1Q2Q3Q4 ~ R1R2R3R4 with perspector P1
Q1Q2Q3Q4 ~ R2R1R4R3 with perspector P2
Q1Q2Q3Q4 ~ R3R4R1R2 with perspector P3
Q1Q2Q3Q4 ~ R4R3R2R1 with perspector P4
Wrt pair of Quadrangles Q1Q2Q3Q4 and P1P2P3P4 (points in given order):
R1R2R3R4 ~ P1P2P3P4 with perspector Q1
R1R2R3R4 ~ P2P1P4P3 with perspector Q2
R1R2R3R4 ~ P3P4R1P2 with perspector Q3
R1R2R3R4 ~ P4P3P2P1 with perspector Q4
“~” stands for “is perspective with”.
See Ref-13, keyword "desmic mate".

Implications
The most important implication of above definition is that any Component Triangle of any given Quadrangle will be perspective with any Component Triangle of any of the other two Quadrangles. Moreover the perspector of any set of two Component Triangles from different Quadrangles will be a vertex of the third Quadrangle.

Desmic Cubic
An extra property is that there is an involutary QA-DT-cubic (see QA-Cu-1) passing through the 3x4 vertices with a pivot point that can be constructed form the 3x4 vertices (method Ref-11, Hyacinthos # 2451).
This cubic was mentioned earlier by Barry Wolk at Ref-11, Hyacinthos # 462.
For QA-examples see Ref-34, QFG#2017.

Different configurations being Desmic
There are configurations with far less elements that can lead to a Desmic Configuration.
A. When a triangle A1B1C1 is perspective with all component triangles of a quadrangle A2B2C2D2, they also form a Desmic Configuration.
They have to be entangled like this:
A1B1C1 is perspective with A2B2C2 with perspector D3
A1B1C1 is perspective with D2C2B2 with perspector A3
A1B1C1 is perspective with C2D2A2 with perspector B3
A1B1C1 is perspective with B2A2D2 with perspector C3
Moreover there will be a perspector D1 of triangles A2B2C2 and A3B3C3.
Now the triple of Quadrangles A1B1C1D1, A2B2C2D2, A3B3C3D3 form a Desmic Configuration.
When the triangle is a Triple Triangle of Px (some point QG-Px or QL-Px) and it is perspective with all Component Triangles of Reference Quadrangle P1.P2.P3.P4, then involved Triple Triangles are called “quadri-perspective” (just like the notions of quadri-orthologic Triple Triangles and quadri-cyclologic Triple Triangles) and the organization is as follows:
Let QG2=P1.P3.P2.P4, QG3=P1.P2.P3.P4 and QG4=P1.P2.P4.P3 be the 3 Component Quadrigons of a Quadrangle (note that the Quadrigon number is determined by the vertex number opposite to P1).
Let Q2 = Px wrt QG2, Q3 = Px wrt QG3, Q4 = Px wrt QG4. Now
Q2Q3Q4 is perspective with P2P3P4 with perspector R1
Q2Q3Q4 is perspective with P1P4P3 with perspector R2
Q2Q3Q4 is perspective with P4P1P2 with perspector R3
Q2Q3Q4 is perspective with P3P2P1 with perspector R4
Moreover there will be a perspector Q1 of triangles P2P3P4 and R2R3R4.
This completes the construction of the triple of Desmic Quadrangles P1P2P3P4, Q1Q2Q3Q4, R1R2R3R4.
Examples:
The Triple Triangles of QG-P12 and QL-P1 are perspective with all 4 Component Triangles of Reference Quadrangle P1.P2.P3.P4 and form a Desmic Configuration.
See further table at QA-Tr-2.

B. For a set of 3 triangles A1B1C1, A2B2C2, A3B3C3 being “Desmic” they have to be entangled like this:
A1 = B2C3∩B3C2, B1 = A2C3∩A3C2, C1 = B2A3∩B3A2
A2 = B3C1∩B1C3, B2 = A3C1∩A1C3, C2 = B3A1∩B1A3
A3 = B1C2∩B2C1, B3 = A1C2∩A2C1, C3 = B1A2∩B2A1.
Consequently the triangles A1B1C1, A2B2C2, A3B3C3 are pairwise perspective.
Let D1 be the perspector of triangles A2B2C2, A3B3C3.
Let D2 be the perspector of triangles A3B3C3, A1B1C1.
Let D3 be the perspector of triangles A1B1C1, A2B2C2.
Now the Quadrangles A1B1C1D1, A2B2C2D2, A3B3C3D3 form a Desmic Configuration.
But also the 3 Triangles being entangled like described before are said to form a Desmic Configuration (because they define the 3 quadrangles).
Examples:
The Triple Triangles of QG-P2, QG-P8, QG-P15 form combined a Desmic Configuration.
The Triple Triangles of QG-P1, QG-P18, QG-P19 form combined a Desmic Configuration.
See Ref-34, QFG #986,#2017.

C. Furthermore even a set of 2 perspective triangles A1B1C1, A2B2C2 is enough to construct a Desmic Configuration:
• Let A3 = B1C2∩B2C1, B3 = A1C2∩A2C1, C3 = B1A2∩B2A1
• Let D1 be the perspector of triangles A2B2C2, A3B3C3.
• Let D2 be the perspector of triangles A3B3C3, A1B1C1.
• Let D3 be the perspector of triangles A1B1C1, A2B2C2.
Now the Quadrangles A1B1C1D1, A2B2C2D2, A3B3C3D3 form a Desmic Configuration.
Examples: See Ref-34, QFG#2017.

There is a special type of Triple Triangle with the property that it has an Orthologic relation with all Component Triangles of the Reference Quadrangle.
It occurs for the Triple Triangles created by QG-P1, QG-P5, QG-P7, QL-P2 and probably more Quadri-points. See QA-Tr-3 and Ref-34, QFG #964, # 980 and #982.
Two triangles A1B1C1 and A2B2C2 are Orthologic if the perpendiculars from the vertices A1, B1, C1 on the sides B2C2, A2C2, and A2B2 are concurrent.
The point of concurrence is known as the Orthology Center of A1B1C1 with respect to A2B2C2.
If this is the case, then the perpendiculars from the vertices A2, B2, C2 on the sides B1C1, A1C1, and A1B1 are also concurrent, as shown by Steiner in 1827. See Ref-13, Orthologic Triangles.
The point of concurrence is known as the Orthology Center of A2B2C2 with respect to A1B1C1.

There is a special type of Triple Triangle with the property that it has a Cyclologic relation with all Component Triangles of the Reference Quadrangle.
The name “Cyclologic” comes from Seiichi Kirikami and Antreas Hatzipolakis.
See QA-Tr-4 and Ref-11, Hyacinthos #22721.
It occurs for the Triple Triangle created by QL-P1 and probably for more Quadri-points. See Ref-34, QFG # 977.
Two triangles A1B1C1 and A2B2C2 are Cyclologic if the A1B2C2, B1A2C2, and C1A2B2 are concurrent in a common point. The point of concurrence is known as the Cyclologic center of A1B1C1 with respect to A2B2C2.
In constrast to the orthologic relation this Cyclologic relation isn’t always reciprocal. However there are many cases that also A2B2C2 will be Cyclologic wrt A1B1C1.

There is a special type of Triple Triangle with the property that it has a Eulerologic relation with all Component Triangles of the Reference Quadrangle. See QA-Tr-6 and Ref-11, Hyacinthos #24415.
It occurs for the Triple Triangle created by QG-P5. See Ref-34, QFG # 2015.
Triangle A1B1C1 is Eulerologic wrt A2B2C2 if the Eulerlines of triangles A2B2C1, A2B1C2, A1B2C2  are concurrent.
The point of concurrence is known as the Eulerologic Center of A1B1C1 with respect to A2B2C2.
In constrast to the orthologic and cyclologic relation this Eulerologic relation isn’t always reciprocal. However there are cases that also A2B2C2 will be Eulerologic wrt A1B1C1.

Vernieuwen