QL-4Tr1: QL-Component Triangles

 
A Quadrilateral has 4 defining random lines L1, L2, L3, L4 without preference or order.
With these lines 4 sets of 3 different lines can be formed:
*     L1, L2, L3
*     L1, L2, L4
*     L1, L3, L4
*     L2, L3, L4
 
Ordened Component Triangles
These 4 sets define 4 triangles also called the Component Triangles of a Quadrilateral.
In certain cases the order within these triangles are of importance.
Well known is that in a Quadrilateral all Component Triangles are perspective with the QL-Diagonal Triangle. However the vertices of the Component Triangles are specific and ordened wrt the QL-Diagonal Triangle.
First of all we have to define the lines of the QL-Diagonal Triangle in an ordened way.
The lines of the QL-Diagonal Triangle can be seen as the connecting lines of the intersection points of opposite lines in the 3 Component Quadrigons (See QL-3QG1).
These 3 Component Quadrigons are:
Note that the serial number of the Sideline of the Diagonal Triangle corresponds to the serial number of the line opposite to L1 in the Quadrigon.
Now Ld2.Ld3.Ld4 is the QL-Diagonal Triangle QL-Tr1.
 
QL-4Tr1-ComponentTriangles-01
 
Defining the lines of the QL-Diagonal Triangle this way we can see in a picture that Ld2.Ld3.Ld4 is perspective with L2.L3.L4.
The other Component Triangles to be perspective with the QL-Diagonal Triangle are L1.L4.L3, L4.L1.L2, L3.L2.L1.
Now the Component Triangles with vertices in the right order to be perspective with QL-Diagonal Triangle:
*     L2  .  L3  .  L4,
*     L1  .  L4  .  L3,
*     L4  .  L1  .  L2,
*     L3  .  L2  .  L1.
The order of reference lines of the QL-Component Triangles also will be of importance when comparing with other triangles built also from QL-Quadrigons. These triangles are also called QL-Triple Triangles (see QL-Tr-1).
The types of relations between a QL-Triple Triangle and simultaneously the 4 QL-Component Triangles are called:
*     Quadri-Perspective relation (examples see QL-Tr-2)
*     Quadri-Orthologic relation (examples see QL-Tr-3)
*     Quadri-Cyclologic relation (examples see QL-Tr-4).
 
 

 

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