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.

 
QA-P13-NinepointCenter-DT-00           

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) (3p+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:
                 S2 + SB SC

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)