QG-2P4: Intersection points QG-Ci1 and QG-Ci2

 
QG-2P4a and QG-2P4b are the intersection points of QG-Ci1 and QG-Ci2.
QG-Ci1 is the circle with the line segment of the side of the QA-Diagonal Triangle opposite the Diagonal Crosspoint QG-P1 as diameter.
QG-Ci2 is the circle with the line segment of the side of the QL-Diagonal Triangle opposite the Diagonal Crosspoint QG-P1 as diameter.
These points are the only finite points (not on the sidelines of the Quadrigon) that can function as perspector for transforming the Reference Quadrigon into a Square. The projective transformation of a Quadrigon into a square is described in Ref-34, QFG#1240.
 
 
 QG-2P4-Intersectionpoints QG-Ci1 and QG-Ci2-01
 
Coordinates:   
CT-coordinates QG-4P1a/b in 1st QA-Quadrigon:
(-2 (a2 - b2 + c2) (-c2 p2 q2 + a2 p2 q r - b2 p2 q r - c2 p2 q r + a2 p q2 r - b2 p q2 r + c2 p q2 r - b2 p2 r2 - a2 p q r2 - b2 p q r2 + c2 p q r2 - a2 q2 r2) + 2 (2 c2 p3 q + 2 c2 p2 q2 - a2 p3 r + b2 p3 r + c2 p3 r - 2 a2 p2 q r + 2 b2 p2 q r - 2 b2 p q r2 + 2 c2 p q r2 - 2 a2 q2 r2 - a2 p r3 - b2 p r3 + c2 p r3 - 2 a2 q r3) S :
-2 a2 c2 p2 q2 + 2 b2 c2 p2 q2 + 2 c4 p2 q2 + a4 p2 q r - 2 a2 b2 p2 q r + b4 p2 q r - 2 a2 c2 p2 q r + 6 b2 c2 p2 q r + c4 p2 q r + 4 a2 c2 p q2 r + 4 b2 c2 p q2 r - 4 c4 p q2 r - 2 a4 q3 r + 4 a2 b2 q3 r - 2 b4 q3 r + 4 a2 c2 q3 r + 4 b2 c2 q3 r - 2 c4 q3 r - 2 a2 b2 p2 r2 + 2 b4 p2 r2 + 2 b2 c2 p2 r2 - 2 a4 p q r2 + 2 b4 p q r2 + 4 a2 c2 p q r2 - 2 c4 p q r2 - 4 a4 q2 r2 + 6 a2 b2 q2 r2 - 2 b4 q2 r2 + 6 a2 c2 q2 r2 + 4 b2 c2 q2 r2 - 2 c4 q2 r2 - a4 q r3 + 2 a2 b2 q r3 - b4 q r3 + 2 a2 c2 q r3 + 2 b2 c2 q r3 - c4 q r3 - 2 (-2 c2 p2 q2 - 4 c2 p q3 + a2 p2 q r - b2 p2 q r - c2 p2 q r + 4 a2 p q2 r - 4 b2 p q2 r - 4 c2 p q2 r + 2 a2 q3 r - 2 b2 q3 r + 2 c2 q3 r + 2 a2 p q r2 - 4 b2 p q r2 - 2 c2 p q r2 - 2 b2 q2 r2 + 2 c2 q2 r2 - 2 b2 p r3 - a2 q r3 - b2 q r3 + c2 q r3) S :
-4 c4 p2 q2 + 4 a2 c2 p2 q r - 4 b2 c2 p2 q r - 4 c4 p2 q r + 4 a2 c2 p q2 r - 4 b2 c2 p q2 r + 4 c4 p q2 r - a4 p2 r2 + 2 a2 b2 p2 r2 - b4 p2 r2 + 2 a2 c2 p2 r2 - 2 b2 c2 p2 r2 - c4 p2 r2 - 2 a4 p q r2 + 4 a2 b2 p q r2 - 2 b4 p q r2 + 2 c4 p q r2 - 2 a4 q2 r2 + 4 a2 b2 q2 r2 - 2 b4 q2 r2 + 4 b2 c2 q2 r2 - 2 c4 q2 r2 - 2 a4 q r3 + 4 a2 b2 q r3 - 2 b4 q r3 + 4 a2 c2 q r3 + 4 b2 c2 q r3 - 2 c4 q r3 - a4 r4 + 2 a2 b2 r4 - b4 r4 + 2 a2 c2 r4 + 2 b2 c2 r4 - c4 r4)
The coordinates of the 2nd point can be derived by changing the sign in the terms with S.
 
Because of the length of the coordinates no other coordinates are shown.
 
 
Properties:
  • When the Reference Quadrigon is line perspective with a Square, then QG-2P4a and QG-2P4b are the only finite points, not on the sidelines of the Reference Quadrigon, that are perspector in this situation. See notes QG-2Cu1.
  • The tangents at QG-2P4a (as well as QG-2P4b) to the circles QG-Ci1 and QG-Ci2 are mutually perpendicular. So these circles are orthogonal circles.
  • The Line through QG-2P4a and QG-2P4b is the Polar (see Ref-13, Polar) of QG-P2 wrt QG-Ci2 and of QG-P3 wrt QG-Ci1.

 

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