QA-L5: Euler Line of QA-Diagonal Triangle
The QA-L5-Line is the Euler Line of the QA-Diagonal Triangle (DT).
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).
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