QA-L5: Euler Line of QA-Diagonal Triangle


The QA-L5-Line is the Euler Line of the QA-Diagonal Triangle (DT).
QA-L5--DT-EulerLine-00         
Coefficients: 
1st CT-coefficient:
–a4 q2 r2 (p + q)2 (p + r)2 (q - r)
–b4 p r2 (p + q)2 (q + r)   (p q2 + q2 r - q r2 + 2 p2 r + 4 p r2 + p q r)
+c4 p q2 (p + r)2 (q + r)   (p r2 + q r2 - q2 r + 2 p2 q + 4 p q2 + p q r)
+a2 b2 q r2 (p + q)2 (3 p2 r2 + 3 q2 r2 + 2 p2 q2 + 2 p3 r + 3 q r3 - p r3 + p2 q r + 3 p q2 r)
–a2 c2 q2 r (p + r)2  (3 p2 q2 + 3 q2 r2 + 2 p2 r2 + 2 p3 q + 3 q3 r - p q3 + p2 q r + 3 p q r2)
+b2 c2 p q r (q2–r2) (2 p4 -p2 q2 -p2 r2 +2 q2 r2 +3 p3 q + 3 p3 r + 2 p2 q r - p q2 r - p q r2)
1st DT-coefficient:
               SA (SB-SC)
Properties:
  • QA-L5 passes through QA-P10 (Centroid DT), QA-P11 (Circumcenter DT), QA-P12 (Orthocenter DT) and QA-P13 (Nine-point Center DT).
  • The Isotomic Centers (QA-P5) of the Centroid Quadrangle, Circumcenter Quadrangle, Orthocenter Quadrangle and Nine-point Center Quadrangle lie on QA-L5, resp. at QA-P10, QA-P12, the De Longchamps Point (ETC-center X(20)) of the QA-Diagonal Triangle and QA-P14 (Eckart Schmidt, August 24, 2012).