QA-P14: Centroid of the Morley Triangle


The QL-Morley Points (QL-P2) of the 3 Quadrigons of the Reference Quadrangle form a triangle Mo1.Mo2.Mo3.
The QL-Quasi Ortholines (see paragraph QL-L6: Quasi Ortholine) of the 3 Quadrigons of the Reference Quadrangle pass through Mo1, Mo2, Mo3 and happen to be the medians of the Morley Triangle.
Their common intersection point is the QA-Quasi Ortholine Point.
This point is also the Centroid of the Morley Triangle.
 QA-P14-CentroidMorleyTriangle-00
                       
Coordinates: 
1st CT-coordinate:
a2 SA Ta - (b2 SB + c2 SC) Tbc + (b2 SB - c2 SC) p (q - r) (q + r)2  +  2 a2 b2 (p - q) (p + q)2 r + 2 a2 c2 q (p - r) (p + r)2
where:
Ta = (3 p2 q2 + 3 p q3 + 2 p2 q r + 9 p q2 r + 5 q3 r + 3 p2 r2 + 9 p q r2 + 6 q2 r2 + 3 p r3 + 5 q r3)
Tbc = (q + r) (6 p3 + 9 p2 q + 9 p2 r + 4 p q2 + 4 p r2 + 3 q2 r + 3 q r2 + 10 p q r)

1st DT-coordinate:
-2 S2 p4 - SB SC (-p2 + q2 + r2)2 + 2 (S2 + 2 SB c2) p2 q2 + 2 (S2 + 2 SC b2) p2 r2 - 4 (a4 - SB SC) q2 r2

Properties:
  • QA-P14 lies on these QA-lines:
        QA-P1.QA-P24           (1 : 2)
        QA-P12.QA-P33         (2 : 1)
  • QA-P14 divides QA-P12.QA-P33 in line segments with ratio 2:1 (QA-P33 = Complement of QA - P12 wrt the Morley Triangle).
  • QA-P14 divides QA-P24.QA-P1    in line segments with ratio 2:1 (QA-P24 = AntiComplement of QA - P12 wrt the Morley Triangle).
  • QA-P14 is the common point of the 3 QA-versions of QL-L6 (Quasi Ortho line).
  • The 3 QL-versions of QA-P14 are collinear and lie on line QL-L6 (note Eckart Schmidt).
  • QA-P14 is the Centroid of Triangle QA-P12.QA-P24.QA-P37.
  • QA-P14 is the Centroid of the Triangle formed by the 3 QA-versions of QL-P2.
  • QA-P14 is the Centroid of the Triangle formed by the 3 QA-versions of QG-P10 (note Eckart Schmidt).
  • QA-P14 is the Perspector of the mutual Triple Triangles (see QA-Tr-1) of QG-P10, QL-P2, QL-P10.

 


Plaats reactie