QL-12L1: A dozen of Equidistance Lines

 
There is the general question of constructing a line in a quadrilateral (a system of 4 lines) such that this line is divided by the 4 lines in 3 equal parts.
This problem is dealt with at Ref-14.
Also a simple synthetic solution is given.
Here the 12 algebraically solutions will be given in QL-notation.
Next picture illustrates how these solutions look like in a square-situation.
 
                        
QL-12L-EquiDistanceLines-00-square
 
 
 
All dotted lines are Equidistance Lines and are divided by the 4 bounding lines of the square in equal parts.
 
When choosing random basic lines for the Reference Quadrilateral it looks like this:
 
 
 QL-12L-EquiDistanceLines-02
 
 
We can split up the dozen equidistance lines per Quadrigon of the Quadrilateral.
In each Auadrigon 4 equidistance lines occur where each middle splitted part of the equidistance line is a line segment between 2 opposite lines of the Auadrigon.
In next figure quadrigon L1.L2.L3.L4 is shown with its 4 equidistance lines.
 
 
Accordingly each equidistance line can be ascribed to 1 of the 3 component quadrigons:
 
Quadrigon                 Quadrigon                 Quadrigon
L1.L2.L3.L4               L1.L3.L2.L4               L1.L2.L4.L3
Equidistance Line          L4132                        L3124                        L2143
Equidistance Line          L2134                        L4123                        L3142
Equidistance Line          L3241                        L1342                        L1234
Equidistance Line          L1243                        L2341                        L4231
Note 1 :
L4132 indicates the equidistance line intersecting QL-lines in order L4, L1, L3, L2.
Note 2 :
The two middle line numbers per equidistance line in the table are opposite lines in the quadrigons in which they occur.
 
 
Coefficients of the 12 lines as well as their 12 Midpoints (see also QL-12P1):
 
The infinity points of the Equidistance Lines (indicating their direction) are important in this setting and have simple coordinates. These coordinates are actually the building blocks of the Equidistance Line itself and its Midpoint as will be shown in next tables.
Next all mentioned coordinates will be CT-coordinates.
 
EquiDistance Lines in Quadrigon L1.L2.L3.L4:
 
                                                         L2134                        L4132                       L1243                       L3241
Infinity point EquiDist.Line      (x : y : z)                   (x : y : z)                   (x : y : z)                   (x : y : z)
                                                          where:                      where:                      where:                      where:
                                                          x = 3 m – n              x = 3 m – 2 n          x = –m – n              x = + m –2 n
                                                          y = n – 2 l                y = 2n – l                 y = + n + 2 l             y = + 2 n + l
                                                          z = 2 l – 3m             z = l – 3m                z = – 2 l + m            z = – l – m
EquiDistance Line                        (2/x : 1/y : 1/z)     (1/x : 1/y : 2/z)    (2/x : 3/y : 1/z)    (1/x : 3/y : 2/z)
EquiDistance Midpoint               (x : 3y : –z)              (x : –3y : –z)           (x : y/3 : –z)            (x : –y/3 : –z)
 
EquiDistance Lines in Quadrigon L1.L3.L2.L4:
 
                                                         L3124                        L4123                      L2341                        L1342
Infinity point EquiDist.Line      (x : y : z)                   (x : y : z)                  (x : y : z)                    (x : y : z)
                                                          where:                     where:                      where:                       where:
                                                          x = m – 3 n             x = 2 m – 3 n           x = 2 m – n              x = –m – n
                                                          y = –3 n +2 l           y = 3 n – l                y = n + l                    y = n – 2 l
                                                          z = –2 l + m            z = l – 2m                z = –l – 2 m             z = 2 l + m
EquiDistance Line                        (-2/x : 1/y : 1/z)     (1/x : –2/y : 1/z)    (1/x : 2/y : –3/z)    (2/x : 1/y : 3/z)
EquiDistance Midpoint               (x : –y : 3 z)            (x : –y : –3 z)          (x : –y : –z/3)          (x : –y : z/3)
 
EquiDistance Lines in Quadrigon L1.L2.L4.L3:
 
                                                         L2143                       L3142                       L1234                       L4231
Infinity point EquiDist.Line      (x : y : z)                  (x : y : z)                   (x : y : z)                  (x : y : z)
                                                         where:                      where:                      where:                     where:
                                                         x = 2 m + n              x = m + 2 n             x = 2 m – n             x = –m – 2 n
                                                         y = –n – l                 y = –2 n + l             y = n – 3 l                y = 2 n – 3 l
                                                         z = l – 2 m               z = –l – m                z = 3 l – 2 m           z = 3 l – m
EquiDistance Line                       (-3/x : 2/y : 1/z)     (–3/x : 1/y : 2/z)    (1/x : -2/y : 1/z)     (1/x : 1/y : 2/z)
EquiDistance Midpoint              (x/3 : y : –z)            (–x/3 : y : –z)         (3 x : y : –z)             (–3 x : y : –z)
 

 

Properties:
  • The Involutary Conjugate (QA-Tf2) of the Infinity Point of an Equidistance Line (performed in the Quadrigon where the Equidistance Line is constructed) coincides with the Equidistance Midpoint (see QL-12P1) on this line.
  • The center of the Circumscribed Hyperbola through P1, P2, P3, P4 and the Infinity Point of an Equidistance Line is the 2nd intersection point of this Equidistance Line and the Nine-point Conic (QA-Co1) of the Reference Quadrigon.

 

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