QA-Ci1: QA-Circumcircle Diagonal Triangle

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The diagonal circle is the circumcircle of the Diagonal Triangle (DT) of a Quadrangle.

The Diagonal Triangle of a Quadrangle is the triangle formed by the intersection points of the 3 distinctive pairs of lines of a Quadrangle.

 

Equations:       

Equation CT-notation:

     a² q r (p+q) (r+p) (+q r x² – p r y² – p q z² – r (p–q) x y + p (q+r) y z + q (r–p) z x)

+ b² p r (p+q) (q+r) (–q r x² + p r y² – p q z² + r (p–q) x y – p (q–r) y z + q (r+p) z x)

+ c² p q (r+p) (q+r) (–q r x² – p r y² + p q z² + r (p+q) x y + p (q–r) y z – q (r–p) z x) = 0

Equation DT-notation:

            a² y z + b² x z + c² x y = 0

Properties:

• QA-P11 is the center of the QA-Ci1.
• QA-P2 (Euler-Poncelet Point) and QA-P30 (Reflection of QA-P2 in QA-P11) lie on QA-Ci1.
• QA-Ci1 is the Involutary Conjugate (see QA-Tf2) of QA-L9, which is the perpendicular bisector of QA-P2.QA-P4.
• The QA-DT-Conic Perspector (see QA-Co-1) of QA-Ci1 is QA-P38.
• Let O1, O2, O3, O4 be the circumcenters of the Component Triangles of Reference Quadrangle P1.P2.P3.P4. Let OPi be the Orthopole (see Ref-13) of line QA-P12.Oi wrt Component Triangle Pj.Pk.Pl, where (i,j,k,l) ∈ (1,2,3,4). The 4 Orthopoles OP1, OP2, OP3, OP4 lie on QA-Ci1. See Ref-11, Hyacinthos messages # 21865 & 21867.
• The QL-Tf1 image of QA-Ci1 is the circle through QG-2P2 a, QG-2P2 b, QG-P19, QA-P4. See Ref-34, Eckart Schmidt, QFG#665.
  
  

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