QG-L6: Inscribed Square Axis


QG-L6 is the line connecting the centers of the 2 possible Inscribed Squares in a Quadrigon.
This line is also mentioned at QA-P23.
 QG-2P1-InscribedSquareCenters-01
       
Coefficients:
CT-Coefficients QG-L6 in 1st QA-Quadrigon:
(2 SB q3 r - 2 SC p q r2 + c2 p q2 (p + 2 q + 2 r) - (b2 - c2) q2 r2 - b2 p2 r2   :
2 SC p q r2 + a2 q2 r2 + p2 (-c2 q2 - 2 SA q r + (a2 - c2) r2)   :
2 SA p2 q r - 2 SB p q3 - a2 q2 r (2 p + 2 q + r) - (a2 - b2) p2 q2 + b2 p2 r2)
CT-Coefficients QG-L6 in 1st QL-Quadrigon:
(-2 SA l m n2 - 2 SB l2 n2 + 2 SC l3 m - (b2 - c2) l3 n + c2 l n3 :
-4 SA m2 n2 + 4 SC l2 m2 + a2 l3 (-2 m + n) - c2 n3 (-2 m + l) - 2 (a2 - b2) l m n2 - 2 (b2 - c2) l2 m n :
-2 SA m n3 + 2 SC l2 m n + 2 SB l2 n2 - a2 l3 n - (a2 - b2) l n3 )
DT-Coefficients QG-L6 in 1st QA-Quadrigon:
            (1   :   (-p2 SA + r2 SC) / (q2 SB)   :   -1)
DT-Coefficients QG-L6 in 1st QL-Quadrigon:  
            (l2 :  2 l m2 n (n4 SA + l4 SC – l n b2 (l2 – l n – n2))
                                              / ((l2 + n2) ( -l2 + 2 l n + n2)(n2 SA + l2 SC) – (l2 – n2)3 SB)  :  n2)
 
Properties:
  • QG-P2 and QA-P23 lie on QG-L6.
  • QG-P2 and QA-P23 lie with the centers of the inscribed squares in harmonic position (Eckart Schmidt, September 6, 2012).
  • The centers of the Inscribed Squares lie center symmetric on a conic which is the QA-Tf2-mapping of the circle with diameter the vertices on the baseline of the QA-Diagonal Triangle. The tangents at the centers of the Inscribed Squares to this conic are parallel to QG-L1 (Eckart Schmidt, September 6, 2012).
  • QG-L6 is the Trilinear Polar of a point TL on QG-L2 wrt the QL-Diagonal Triangle QL-Tr1 (the trilinear polar of P is the perspectrix of the Cevian triangle of P and the Reference Triangle, see Ref-13) (Eckart Schmidt, September 6, 2012).
  • QG-L6 is the Trilinear Polar of a point TA on QG-P1.QG-P14 (line parallel to QG-L1) wrt the QA-Diagonal Triangle (QA-Tr1) (Eckart Schmidt, September 6, 2012).
 QG-L6-Inscribed Square Axis-10

  • The Trilinear Poles TA und TL of QG-L6 wrt QA-DT und QL-DT are collinear with QG-P2 (Eckart Schmidt, September 6, 2012).

 


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