QG-5: List of QG-Crosspoints

 
In next list sets of 3 lines are mentioned that concur.
All lines with points in the range QG-P1QG-P14, QA-P1 - QA-P21 and QL-P1 - QL-P23 have been taken into account.
Lines are defined by 2 points on it with lowest serial number.
There are regularly recurring crossing lines with these Crosspoints. This is an indication for the occurrence of Perspective Fields (see QA-PF-1).
When the intersection points have fixed ratios of the distances to the defining points on the defining lines, then they are mentioned. There are many of them.
When there are no fixed ratios this is indicated by the remark “x : y”.
For point P on line P1.P2 the ratio d1 : d2 means that d(P,P1) : d(P,P2) = d1 : d2, where:
  • d1 is positive when P is positioned wrt P1 at the same side of the line as P2. If not then d1 is negative.
  • d2 is positive when P is positioned wrt P2 at the same side of the line as P1. If not then d1 is negative.

 

lines                                                                                      ------ratios-----        comment    

QG-P1.QG-P2   ^ QG-P10.QA-P14 ^ QG-P13.QA-P2        x:y     x:y     x:y
QG-P1.QG-P2   ^ QG-P12.QG-P14 ^ QL-P18.QL-P23      x:y     x:y     x:y           Point on QA-Co1 (Nine-point Conic)
QG-P1.QG-P4   ^ QG-P5.QG-P11   ^ QG-P6.QG-P9          3:2    4:1    4:1            (simple coordinates)
QG-P1.QG-P5   ^ QG-P7.QG-P10   ^ QG-P11.QA-P1                                               Infinity Point
QG-P1.QG-P6   ^ QG-P5.QG-P8     ^ QG-P11.QA-P1          1:3    3:-1   2:-1          Point on QA-Co1 (Nine-point Conic)    
QG-P1.QG-P6   ^ QG-P7.QG-P9     ^ QL-P2.QL-P7                                               Infinity Point Steiner Line
QG-P1.QG-P7   ^ QG-P4.QG-P10   ^ QG-P6.QA-P1           4:1    2:3     1:4
QG-P1.QG-P7   ^ QG-P5.QG-P10   ^ QG-P6.QG-P9         2:1    2:1      2:1          Centroid QG-P7.P9.P10+QG-P1.P5.P6
QG-P1.QG-P14 ^ QG-P2.QG-P3     ^ QG-P6.QL-P18                                            Infinity Point (simple coords)
QG-P1.QA-P5   ^ QG-P2.QA-P1      ^ QG-P8.QA-P20       1:1    2:1    -1:3           Point on Newton Line
QG-P1.QA-P5   ^ QG-P2.QA-P20   ^ QG-P4.QA-P10                                            Infinity Point
QG-P1.QA-P17  ^ QG-P4.QG-P12    ^ QG-P14.QA-P5        x:y   -2:3     x:y
QG-P1.QA-P19 ^ QG-P2.QG-P3     ^ QG-P14.QA-P5        x:y     x:y     x:y          Point on QA-Cu4.
QG-P1.QA-P19 ^ QG-P2.QG-P12   ^ QG-P14.QA-P10      x:y     x:y     3:1
QG-P1.QA-P20 ^ QG-P2.QG-P3    ^ QG-P12.QA-P18      x:y     x:y     x:y               
                                                               ^ QG-P14.QA-P10                           3:1  
QG-P1.QA-P20 ^ QG-P2.QG-P12   ^ QG-P8.QA-P10                                           Point on QA-Cu2, and
                                                               ^ QG-P14.QA-P5                                            Infinity Point Newton Line             
QG-P1.QL-P9    ^ QG-P3.QL-P10   ^ QG-P5.QL-P17        2:-1   -1:2    x:y          Point on QL-Ci1 (DT-circumcircle)
QG-P1.QL-P10  ^ QG-P3.QL-P9     ^ QG-P7.QL-P2                                              Infinity Point (simple coords)
QG-P1.QL-P12  ^ QG-P2.QG-P4     ^ QG-P3.QL-P18       6:-1    3:2    3:2
QG-P2.QG-P4   ^ QG-P8.QA-P10    ^ QL-P12.QA-P25    -1:2     3:-2   -3:4        Only (Quasi-)Centroids involved
QG-P2.QG-P5   ^ QG-P7.QA-P5      ^ QG-P9.QA-P10      -1:2    -1:2    3:-2
QG-P2.QG-P6   ^ QG-P5.QA-P5      ^ QG-P10.QA-P10    -1:2    -1:2    3:-2
QG-P2.QG-P7   ^ QG-P5.QA-P5      ^ QG-P10.QA-P20                                          Infinity Point
QG-P3.QG-P7   ^ QG-P5.QL-P9     ^ QG-P10.QL-P11       x:y     x:y     x:y
QG-P4.QG-P9   ^ QG-P5.QG-P11   ^ QG-P6.QA-P1           4:3    4:3     8:-1
QG-P6.QG-P9   ^ QG-P7.QG-P8    ^ QG-P11.QA-P1          3:1     3:1     1:1
QG-P6. QL-P9   ^ QG-P7.QL-P2    ^ QG-P10.QL-P11       x:y     x:y     x:y

 
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