QA-P10: Centroid of the QA-Diagonal Triangle
The Diagonal Triangle of a Quadrangle P1.P2.P3.P4 is the triangle built from the intersection points S1 = P1.P2 ^ P3.P4, S2 = P1.P3 ^ P2.P4 and S3 = P1.P4 ^ P2.P3.
These points have CT-coordinates: S1 = (p : q : 0), S2 = (p: 0 : r), S3 = (0 : q : r).
Because of the symmetry in S1, S2, S3 all Triangle Centers wrt S1.S2.S3 as described in Clark Kimberling’s Encyclopedia of Triangle Centers (Ref-12) also will be Quadrangle Centers.
However only those points contributing to the points derived from component Quadrigons or component triangles will be described here as Quadrangle Centers.
The Centroid of the Diagonal Triangle does contribute to the points described earlier. The relation with the Isotomic Center QA-P5 is most special.
p (q + r) (2 p + q + r)
- QA-P10 lies on these QA-lines:
QA-P10 lies on these QG-lines:
- QA-P10 is the Reflection of QA-P25 in QA-P26.
- QA-P5.QA-P1 : QA-P1.QA-P10 : QA-P10.QA-P20 = 3 : 1 : 2.
This is the same ratio as used in the construction of the Quadrangle Centroid G using component triangles (when Gi = Centroid Pj.Pk.Pl then Pi.G : G.Gi = 3 : 1).
As a consequence Quadrangle S1.S2.S3.QA-P5 and the Reference Quadrangle P1.P2.P3.P4 share the same Centroid (QA-P1).
- QA-P10 is the Perspector of the Triple Triangles (see QA-Tr-1) of these pairs (QG-P1, QG-P2), (QG-P4, QL-P12).
- QA-P2 is the Parallellologic Center of the QL-P8 Triple Triangle wrt the Triple Triangles of QG-P1/QG-P2/QG-P4/QG-P8/QG-P15 TT. See QA-Tr-1.