QA-P40 Centroid of the QL-P6-Triple Triangle

QA-P40 is the Centroid of the QL-P6-Triple Triangle.

It is also the Centroid of the QG-P9-Triple Triangle.

It is also the Circumcenter of the QL-P17-Triple Triangle.

For the definition of a Px-Triple Triangle see QA-Tr-1.

1st CT-Coordinate

-a⁴ (p + q) (p + r) (4 q² + 6 q r + 4 r² + 5 p (q + r)) + (-b² + c²) p (q + r) (-c² (p + r) (2 p + 3 q + r)

+ b² (p + q) (2 p + q + 3 r)) + a² (c² (p + r) (q (3 p + 2 q)² + (7 p² + 15 p q + 6 q²) r + (5 p + 2 q) r²)

+ b² (p + q) (2 r (q + r) (q + 2 r) + p² (7 q + 9 r) + p (5 q² + 15 q r + 12 r²)))

1st DT-Coordinate

a⁴ (3 p⁴ + q⁴ + 6 q² r² + r⁴ – 4 p² (q² + r²)) + 2 (b – c) (b + c) p² (b² (p² – q² – 3 r²) + c² (-p² + 3 q² + r²))

– a² (b² (5 p⁴ + (q² – r²)² – 2 p² (3 q² + r²)) + c² (5 p⁴ + (q² – r²)² – 2 p² (q² + 3 r²)))

Properties

• QA-P40 lies on these lines:
  • QA-P11.QA-P32                                                                  (2 :  1)
  • QA-P14.QA-P26                                                                  (3 : -1)
  • QG-P9.QL-P6                                                                       (2 :  1)
  • Midpoint(QA-P5,QA-P12) . Midpoint(QA-P1,QA-P32)   (4 : -1)
• QA-P40 is:
  • the Circumcenter of the QL-P17-Triple Triangle
  • the Centroid of the QL-P6-Triple Triangle
  • the Centroid of the QG-P9-Triple Triangle

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