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.

Equation CT-notation:

(x + y + z) (T_MX x + T_MY y + T_MZ z) + 2 T_XYZ (a² y z + b² x z + c² x y) = 0

where:

T_MX = 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)

T_MY = 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)

T_MZ = 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)

T_XYZ = 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 3^rd 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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