QA-P11: Circumcenter of the Diagonal Triangle

QA-P11 is the Circumcenter of the Diagonal Triangle (QA-Tr1) of a Quadrangle. Coordinates:
1st CT-Coordinate:

- a2 q r (2 SA p2 q r + TA) + (SC TB p r + SB TC p q) + 2 S2 p2 q r (q+r) (p+q+r),

where:
TA =  -a2 q2 r2 + b2 p2 r2 + c2 p2 q2
TB = +a2 q2 r2 - b2 p2 r2 + c2 p2 q2
TC = +a2 q2 r2 + b2 p2 r2 - c2 p2 q2

1st DT-Coordinate:
a2 SA

Properties:
• QA-P11 lies on these QA-lines:
QA-P2.QA-P30          (1 :  1 => QA-P11=Midpoint QA-P2.QA-P30)
QA-P10.QA-P12         (1 : 2 => QA-P11=Complement of QA-P12 wrt QA-DT)
QA-P20.QA-P37        (1 :  1 => QA-P11=Midpoint QA-P20.QA-P37)
• QA-P11 point is the center of the circumscribed circle through the vertices of the QA-Diagonal Triangle. This circle is interesting because QA-P2 (Euler-Poncelet Point) is situated on it.
• QA-P11 is the intersection point of the directrices of the 2 parabolas of the Reference Quadrangle.
• QA-P11 is collinear with QA-P10 (Centroid DT), QA-P12 (Orthocenter DT), QA-P13 (Nine-point Center DT) on the Euler Line of the QA-Diagonal Triangle.
• QA-P11 is the Gergonne-Steiner point (QA-P3) as well as the Isogonal Center (QA-P4) from the Quadrangle formed by the vertices of the Diagonal Triangle and QA-P2 (Euler-Poncelet Point).
• QA-P11 is concyclic with the Involutary Conjugate (see QA-Tf2) of QA-P11 and the foci of the QA-Parabolas QA-2Co1.
• QA-P11 is the Isotomic Center (QA-P5) of the Nine-point Center Quadrangle (Eckart Schmidt, August 24, 2012).
• QA-P11 is the QA-Orthopole(QA-Tf3) of QA-P12.
• QA-P11 is the Orthology Center of the QG-P1 Triple Triangle wrt the QG-P17 Triple triangle. See QA-Tr-1.
• QA-P11 is the Orthology Center of the QG-P2 Triple Triangle wrt the Triple triangles of QG-P1/QG-P4/QG-P8/QG-P15/QL-P12. See QA-Tr-1.
• QA-P11 is the Orthology Center of the QG-P8 Triple Triangle wrt the Triple triangles of QG-P5,QG-P10,QL-P2. See QA-Tr-1.
• The reflection of QA-P11 in QA-P1 is the midpoint of QA-P5 and QA-P12.

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