QA-Tr-4: (Quadri-)Cyclologic QA-Triple Triangles

 
Cyclologic pairs of Triple Triangles
Two triangles A1B1C1 and A2B2C2 are Cyclologic if the circles 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.
If this is the case, then the circles A2B1C1, B2A1C1, C2A1B1 also will be concurrent.
The point of concurrence is known as the Cyclologic center of A2B2C2 with respect to A1B1C1.
There is a Quadri Cyclologic relationship when the Triple Triangle is cyclologic with all Component Triangles of the Reference Quadrangle. See QA-Tr-1.
Here is a list of Cyclologic pairs of Triple Triangles in a Quadrangle.
 
 
Triple Triangle-1
formed by 3
QA-versions of:
Triple   Triangle-2
formed by 3
QA-versions of:
Cyclologic Center-1 Cyclologic Center-2
Perspective Center
*)
 
QG-P1 QG-P15 QA-P2 QA-P3 QA-P1
QG-P1 QG-P16
3rd intersection of QA-Cu1 and the line through QA-P3 and the intersection of QA-Cu1 and its
asymptote   *)
QA-Tf2(QA-Cu1 ^ asymptote) *) Infinity Point of QA-Cu1-asymptote
QG-P1 QG-P18 QA-P2  *) QA-P41 QA-Tf2(X)
QG-P1 QG-P19 QA-P2 QA-P4 X on a QA-L2-parallel through QA-P12
QG-P1 QL-P1 QA-P3 QA-P41 QA-P3
QG-P1 QL-P17 QA-Px QA-P3 no
QG-P2 QL-P1 QA-P41  *) QA-P1 no
QG-P3 QL-P17 QA-Px QA-Px no
QG-P4 QG-P8 QA-P34 QA-P1.QA-P2 with ratio 1:2   *) QA-P1
QG-P5 QG-P10 QA-Px QA-Px QA-P1
QG-P5 QL-P1 QA-P3 QA-P9 QA-Px
QG-P5 QL-P4 QA-Px QA-Px no
QG-P6 QL-P21 QA-Px QA-Px no
QG-P7 QG-P9 QA-Px QA-Px QA-P1
QG-P9 QL-P1 QA-P1 QA-P9   *) no
QG-P9 QL-P4
point on
QA-Px no
QG-P9 QL-P5 QA-Px QA-Px no
QG-P12 QL-P1 QA-Px QA-Px no
QG-P16 QL-P1 QA-P4 tangential of  QA-P3 wrt QA-Cu1  *) QA-P4
QG-P18 QG-P19 QA-P41 QA-P4 On X.QA-Tf2(X)
QL-P1 QL-P17
point on circumcircle
QA-Tr2   *)
QA-P3 no
             QL-P1
All Component Triangles
Common
Quadri-Cyclologic
point QA-P4
4 different
Quadri-Cyclologic
Points
4 different
Quadri-Perspective
Points
 
QA-Px is a QA-point not registered in EQF.
Note: if the perspective Center (=perspector) is QA-P1, their cyclologic centers will be the 4th intersection points of the triangle circumcircles and the conic through their 6 vertices.  *)
*) these points/notes were identified by Eckart Schmidt. See Ref-34, QFG-messages #1971, #1983.