QA-P13: Nine-point Center of the QA-Diagonal Triangle

QA-P13 is the Nine-point Center of the Diagonal Triangle (QA-Tr1) of a Quadrangle.
It is also the center of the circumcircle of the Medial Triangle (MT) of the QA-Diagonal Triangle (DT).
The sides of the MT are tangential to both Quadrangle Parabolas.

1st CT-Coordinate:

– a² q r (2 S_A p² q r + T_A) + (S_C T_B p r + S_B T_C p q) – 2 S² p² q r (q+r) (3p+q+r),
where:
T_A = -a² q² r² + b² p² r² + c² p² q²
T_B = +a² q² r² – b² p² r² + c² p² q²
T_C = +a² q² r² + b² p² r² – c² p² q²

1st DT-Coordinate:

S² + S_B S_C

Properties:

• QA-P13 lies on these QA-lines:
  • QA-P10.QA-P11 (-1 : 3)
  • QA-P29.QA-P36 ( 1 : 1 => QA-P13 = Midpoint QA-P29.QA-P36)
  • QA-P30.QA-P35 (5 : -1)
• QA-P13 = Midpoint of QA-P11.QA-P12.
• QA-P13 = the center of QA-Ci2, the circumcircle of the Medial Triangle (MT) of the QA-Diagonal Triangle (DT).
• QA-P13 = QA-centroid (QA-P1) of the quadrangle formed by the vertices of the Diagonal Triangle and QA-P12.
• QA-P13 = QA-Centroid of Quadrangle QA-P2.QA-P3.QA-P12.QA-P20.
• QA-P13 is the Orthology Center of the QG-P2 Triple Triangle wrt the QG-P17 Triple triangle. See QA-Tr-1.

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