QA-L6: QA-Newton-Morley Line


Each QL-line parallel to the Newton Line when transferred to the 3 Quadrigons of a Quadrangle produces 3 concurrent lines.
The locus of the common point is QA-L6.
QA-P1 (the Quadrangle Centroid) is a point on this line because the QL-Newton Line is transferred into QA-P1.
QA-L6-Newton-Morley-Line-00                    
Coefficients:  
1st CT-coefficient:
-a4 q r (p + q) (q - r) (p + r)
-c4 (p + r) (q + r) (p3 + 3 p2 q + 4 p q2 + 2 q3 + 2 p2 r + 3 p q r + 3 q2 r + p r2 + q r2)
+b4 (p + q) (q + r) (p3 + 2 p2 q + p q2 + 3 p2 r + 3 p q r + q2 r + 4 p r2 + 3 q r2 + 2 r3)
+b2 c2 (q - r) (q + r) (-p2 q - p q2 - p2 r + q2 r - p r2 + q r2)
+a2 c2 (p + r) (p3 q + 3 p2 q2 + 4 p q3 + 2 q4 + p3 r + 7 p2 q r + 12 p q2 r + 6 q3 r + 2 p2 r2
                                                                                                      + 7 p q r2 + 7 q2 r2 + p r3 + 3 q r3)
-a2 b2 (p + q) (p3 q + 2 p2 q2 + p q3 + p3 r + 7 p2 q r + 7 p q2 r + 3 q3 r + 3 p2 r2 + 12 p q r2
                                                                                                         + 7 q2 r2 + 4 p r3 + 6 q r3 + 2 r4)
1st DT-coefficient:
            (-c2 q2+b2 r2 (p2 (p4-3 (q2-r2)2+2 p2 (q2+r2)) SA+(3 p4-(q2-r2)2-2 p2 (q2+r2)) (q2 SB+r2 SC))
 
Properties:
  • Since the QL-Morley Line is parallel to the QL-Newton Line also the common QA-point of the Morley lines QA-P15 lies on QA-L6.


 

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