QA-4Tr1: QA-Component Triangles

 
A Quadrangle has 4 defining random points P1, P2, P3, P4 without preference or order.
With these points 4 sets of 3 different points can be formed:
*   P1, P2, P3
*   P1, P2, P4
*   P1, P3, P4
*   P2, P3, P4
 
Ordened Component Triangles
These 4 sets define 4 triangles also called the Component Triangles of a Quadrangle.
In certain cases the order within these triangles are of importance.
Well known is that in a Quadrangle all Component Triangles are perspective with the QA-Diagonal Triangle. However the vertices of the Component Triangles are specific and ordened wrt the QA-Diagonal Triangle.
First of all we have to define the vertices of the QA-Diagonal Triangle in an ordened way.
The vertices of the QA-Diagonal Triangle can be seen as the Diagonal Crosspoints of the 3 Component Quadrigons (See QA-3QG1).
These 3 Component Quadrigons are:
Note that the serial number of the Diagonal Crosspoint corresponds to the serial number of the point opposite to P1 in the Quadrigon.
Now S2.S3.S4 is the QA-Diagonal Triangle QA-Tr1.
 
QA-4Tr1-ComponentTriangles-01
 
Defining the vertices of the QA-Diagonal Triangle this way we can see in a picture that S2.S3.S4 is perspective with P2.P3.P4.
The other Component Triangles to be perspective with the QA-Diagonal Triangle are P1.P4.P3, P4.P1.P2, P3.P2.P1.
Now the Component Triangles with vertices in the right order to be perspective with QA-Diagonal Triangle are:
*   P2  .  P3  .  P4,
*   P1  .  P4  .  P3,
*   P4  .  P1  .  P2,
*   P3  .  P2  .  P1.
The order of vertices of the QA-Component Triangles also will be of importance when comparing with other triangles built also from QA-Quadrigons. These triangles are also called QA-Triple Triangles (see QA-Tr-1).
The types of relations between a QA-Triple Triangle and simultaneously the 4 QA-Component Triangles are called:
*   Quadri-Perspective relation (examples see QA-Tr-2)
*   Quadri-Orthologic relation (examples see QA-Tr-3)
*   Quadri-Cyclologic relation (examples see QA-Tr-4).
 

 

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