QA-Tr-6: Eulerologic QA-Triple Triangles
Eulerologic pairs of Triple Triangles
The notion of two triangles having an Eulerologic relation was introduced by Antreas Hatzipolakis. See Ref-11, Hyacinthos #24415.
Triangle A1B1C1 is Eulerologic wrt A2B2C2 if the Eulerlines of triangles A1B2C2, A2B1C2, A2B2C1 are concurrent. The point of concurrence is known as the Eulerologic Center of A1B1C1 with respect to A2B2C2.
In constrast to the orthologic, cyclologic and parallelologic relationship this Eulerologic relationship isn’t always reciprocal. However there are cases of reciprocity.
Here is a list of Eulerologic pairs of Triple Triangles in a Quadrangle.
| Nr |
Triple Triangle-1
formed by 3
QA-versions of:
|
Triple Triangle-2
formed by 3
QA-versions of:
|
Eulerologic Center-1 | Eulerologic Center-2 |
| 1 | QG-P1 | QG-P2 |
InfinityPoint(QA-L5)
=X(30) of QA-DT
|
= X(3) of QA-DT
|
| 2 | QG-P1 | QG-P17 | X(125) of QA-DT *) |
= X(2) of QA-DT *)
|
| 3 | QG-P2 | QG-P17 |
= X(5) of QA-DT
= X(3) of QG-P17TT
|
= X(5) of QA-DT
= X(3) of QG-P2TT
|
| 4 | QG-P5 | QL-P4 | none | QA-P1.QA-P15 (-1:2) |
| 5 | QG-P5 |
All Component Triangles QA-CTi
|
4 points E1,E2,E3,E4,
where QG-P5 TT is the DT of E1.E2.E3.E4.
Ei=X(3) of QA-CTi.
|
none |
| 6 | QG-P8 | QL-P12 |
InfinityPoint(QA-L5)
=X(30) of QA-DT
|
X(3) of QG-P8TT |
| 7 | QG-P9 | QL-P1 | X(3) of QL-P1TT | none |
| 8 | QG-P9 | QL-P6 | Infinitypoint of the Eulerline of QL-P6TT | X(3) of QG-P9TT |
| 9 | QG-P10 | QL-P2 | Infinitypoint of the Eulerline of QL-P2TT | X(3) of QG-P10TT |
| 10 | QG-P10 | QL-P3 | none | QA-P15 |
| 11 | QG-P10 | QL-P29 | none | QA-P15 |
QG-PiTT/QL-PiTT stands for Triple Triangle of QG-Pi/QL-Pi (i=serial number of point).
QA-DT stands for the Diagonal Triangle of the Reference Quadrangle.
QA-CTi stands for the Component Triangle-i (i=1,2,3,4) of the Reference Quadrangle.
*) see Ref-33, property Seiichi Kirikami at Anapolis #4178
Summary:
- The first 3 mentioned instances of Eulerologic Centers all fully relate to the QA-Diagonal Triangle and therefore are Eulerologic examples wrt a Reference Triangle.
- In instance 2 there is not only an Eulerologic relationship but also a “Brocardologic” relationship because also the Brocard Axes of involved triangles concur. They even concur with the Eulerlines because involved triangles are isosceles.
- Instances 1, 6, 8, 9 relate to triangles with their medial triangle giving parallel Eulerlines, meeting in their Infinitypoint, and Eulerlines meeting in the circumcenter of the larger triangle.
- Instance 5 is an example of a non-recursive Quadri-Eulerologic relationship where a Triple Triangle is Eulerologic related to all QA-Component Triangles. The result is 4 Eulerologic Centers forming a quadrangle with the Triple Triangle as Diagonal Triangle.
- The other instances are specific non-recursive Eulerologic relationships with only one Eulerologic Center.
- It is remarkable that often the Eulerologic Center is the circumcenter of one of the Triple Triangles.