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.
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 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-P1.QA-P4 *) | 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 (QA-4Tr1) | 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 [34], QFG#1971,#1983.
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