QL-P9: Circumcenter QL-Diagonal Triangle

QL-P9 is the Circumcenter of the Diagonal Triangle (QL-Tr1) of the Reference Quadrilateral.

1^st CT-coordinate

4 m n (-4 Δ² m² n² + a² l³ (S_C m + S_B n) – l² (S_C m + S_B n)² + l m n (S_B b² m + S_C c² n))

where:

S_A = (–a² + b² + c²)/ 2 S_B = (+a² – b² + c²)/ 2 S_C = (+a² + b² – c²)/ 2

Δ = Area = 1/4 √[(a + b + c) (-a + b + c) (a – b + c) (a + b – c)]

1^st DT-coordinate

a² Sa

Properties

• QL-P9 lies on these lines:
  • QL-P2.QL-P7  =QL-L2
  • QL-P8.QL-P10=QL-L7        (-1 : 3 => QA-P9 = Complement of QA-P10)
• QL-P9 lies together with QL-P8, QL-P10 and QL-P11 on QL-L7, the QL-Euler Line of the Diagonal Triangle. The ratio of distances between QL-P9, QL-P8, QL-P11, QL-P10 in this order is 2 : 1 : 3.
• QL-P9.QL-P16 // QL-P1.QL-P11.
• QL-P9 lies on the Polar (see [13], Polar) of QL-P25 wrt QL-Co3 and vice versa.
• QL-P9 lies on the circumcircle of the triangle formed by the 3 QL-versions of QG-P9.
• The Clawson-Schmidt Conjugate (QL-Tf1) of QL-P9 lies on QL-Ci3 (Miquel Circle).
• The directrices of QL-Co1 and QL-Co3 meet in QL-P9 (Circumcenter QL-Diagonal Triangle).
• QL-P1 lies on the Polar (see [13], Polar) of QL-P9 wrt QL-Co1.

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