QA-P32: Centroid of the Circumcenter Quadrangle
QA-P32 is the QA-Centroid of the quadrangle formed by the Circumcenters of the Component Triangles of the Reference Quadrangle.
Coordinates:
1st CT-Coordinate:
-a4 q2 r2 + SB c2 p2 q2 + SC b2 p2 r2 + 2 S2 p2 q r + 3 a2 SA p q r (p + q + r)
1st DT-Coordinate:
a4 q2 r2 - S2 p4 + (c2 SC - SB2) p2 q2 + (b2 SB - SC2) p2 r2
Properties:
- QA-P32 lies on these QA-lines:
- QA-P1.QA-P32.QA-P33 // QA-P2.QA-P4.QA-P6 = QA-L2.
- The QA-Orthopole (QA-Tf3) of QA-P32 is intersection point QA-P1.QA-P32 ^ QA-P2.QA-P7.
- QA-P32 is the Orthology Center of the QG-P9 Triple Triangle wrt the Triple triangles of QG-P1/QG-P2/ QG-P4/QG-P8/QG-P15/QL-P12. See QA-Tr-1.
- QA-P32 is the Orthology Center of the QG-P5 Triple Triangle wrt the Triple triangles of QG-P12/QG-P14. See QA-Tr-1.
- Let L12, L34, L13, L24, L14, L23 are the perpendicular bisectors of segments P1P2, P3P4, P1P3, P2P4, P1P4, P2P3 respectively. Then Newton lines of the Quadrilaterals (L12,L34, L13,L24) , (L12,L34, L14,L23) and (L13,L24, L14,L23) are concurrent in QA-P32. See Ref-34, Tran Quang Hung, QFG#3735, #3738.