QA-P43: Least Squared Distance Point QA and QA-DT
This point minimizes the sum of squared distances to these 7 points:
- 4 vertices of the Reference Quadrangle
- 3 vertices of the QA-Diagonal Triangle (QA-Tr1)
QA-P43 is also the Homothetic Center (Perspector) of the QA-Diagonal Triangle (QA-Tr1) and the QL-P12-Triple Triangle (QA-Tr4).
Construction:
There is a construction for this point based on the "12 Midpoints" construction” in a triangle. See Ref-33, Anopolis # 899 and Ref-34, QFG # 598.
Let P be a point; A', B', C' its cevian traces. Let Ab, Ac be the midpoints of AB', AC' and define analogously Bc, Ba, Ca, Cb. Let A1, B1, C1 be the midpoints of AA', BB', CC' and A2, B2, C2 the midpoints of BcCb, CaAc, AbBa. The lines A1A2, B1B2, C1C2 concur.
Applied in a Quadrangle (triangle Pi.Pj.Pk with random point Pl where (i,j,k,l) ∈ (1,2,3,4)) it produces a Quadrangle homothetic with the Reference Quadrangle giving Homothetic Center on QA-P1.QA-P10 (3 : 4).
Coordinates:
1st CT-Coordinate:
(2 p + q + r) (q + r) (2 p2 + 2 p q + 2 p r + q r)
1st DT-Coordinate:
5 p4 + q4 + r4 - 6 p2 q2 - 6 p2 r2 - 2 q2 r2
Properties:
- QA-P43 lies on these lines:
- QA-P43 is the Homothetic Center (Perspector) of the QA-Diagonal Triangle (QA-Tr1) and the QL-P12-Triple Triangle (QA-Tr4). See QA-Tr-1 and QA-Tr-2.
- QA-P43 is the Homothetic Center (Perspector) of the QG-P3-Triple Triangle and the QL-P15-Triple Triangle. See QA-Tr-1 and QA-Tr-2.
- QA-P43 is the Homothetic Center (Perspector) of the QG-P2-Triple Triangle and the medial triangle of the QL-P12-Triple Triangle. See QA-Tr-1.
- QA-P43 is the Homothetic Center (Perspector) of the QG-P4-Triple Triangle and the medial triangle of the QG-P2-Triple Triangle. See QA-Tr-1.