QA-Ci2: QA-Medial Circle Diagonal Triangle


The Diagonal Medial Circle is the circumcircle of the Medial Triangle (MT) of the Diagonal Triangle (DT) of a Quadrangle.
It is also the Nine-point Circle (or also called Euler Circle) of the QA-Diagonal Triangle.
 
QA-Ci2-DiagonalMedialCircle-01

Equations:
 
Equation CT-notation:  
          (x + y + z) (TMX x + TMY y + TMZ z) + 2 TXYZ (a2 y z + b2 x z + c2 x y) = 0
where:
          TMX = qr(a2 qr(p+q)(r+p) - b2 pr(p+q)(q+r) - c2 pq(r+p)(q+r)) - 2 q2r2 (a2 qr+b2 rp+c2 pq)
          TMY = pr(–a2qr(p+q)(r+p)+b2 pr(p+q)(q+r) - c2 pq(r+p)(q+r)) - 2 r2p2 (a2 qr+b2 rp+c2 pq)
          TMZ = pq(–a2qr(p+q)(r+p)-b2pr(p+q)(q+r) + c2 pq(r+p)(q+r)) - 2 p2 q2(a2 qr+b2 rp+c2 pq)
          TXYZ = 2 pqr (p+q) (q+r) (r+p)

Equation DT-notation:
            SA x2 + SB y2 + SC z2 - c2 x y - b2 x z - a2 y z = 0

Properties:

  • These points lie on QA-Ci2:
            Foci of circumscribed Quadrangle Parabolas (QA-2Co1a and QA-2Co1b)
            QA-P29: Complement of QA-P2 wrt the QA-Diagonal Triangle
            QA-P36: Complement of QA-P30 wrt the QA-Diagonal Triangle
            QG-P2: Midpoint 3rd QA-Diagonal (evident property)
            QG-P17: Projection QG-P1 on QG-L1
  • QA-P13 is the center of the Medial Circle.
 


 

 
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