QL-P3: Kantor-Hervey point

 
The circumcenters of the 4 component triangles of the Reference Quadrilateral are concyclic (on the Miquel Circle). These circumcenters Oi (i=1,2,3,4) form a Quadrangle with 4 component triangles whose Orthocenters HOi (i=1,2,3,4) are also concyclic (on the Hervey Circle) with circumcenter QL-P3, the Kantor-Hervey Point.
QL-P3 is also described by F. Morley (1902) as point p1, being the center of the inscribed deltoid of a 4-Line in Ref-49 page 4.
There is a description of this point in Ref-2b, Jean-Louis Ayme Le point de Kantor-Hervey”. Here the point is defined as the common point of the perpendicular bisectors of Oi.Hi (i=1,2,3,4), where Oi and Hi are resp. the Circumcenter and the Orthocenter of the Component Triangles of the Reference Quadrilateral.
There is also a description in Ref-6 by Alain Levelut.
 
 
 
QL-P3-HerveyPoint-00
 
Coordinates
 
1st CT-coordinate:
            a2 (l - m) (l - n) (b2 n - c2 m) + (b2 - c2) (m - n) (-b2 (l - m) n + c2 m (l - n))
 
1st DT-coordinate:
            a4  -  b4 (3 m2+n2)/(l2-n2)  -  c4 (m2+3 n2)/(l2-m2)
- 4 b2 c2 m2/(-l2+m2)  -  4 b2 c2 n2/(-l2+n2)  +  2 a2 (b2-c2) (m2+n2)/(m2-n2)
 
 
Properties:
  • QL-P3 lies on this line:
        QL-P4.QL-P5       (2 : -1 => QL-P3 = Reflection of QL-P4 in QL-P5)
  • Distances QL-P4.QL-P6 : QL-P6.QL-P5 : QL-P5.QL-P3 = 1 : 1 : 2.
  • QL-P2.QL-P3 = QL-L4 (Morley Line) // Newton Line QL-L1.
  • QL-P3 is also the point of concurrence of the four perpendicular bisectors of the segments Oi.Hi of the Euler Lines of the QL-Component Triangles Ti (Ref-6, page 5).
  • QL-P3 is the center of the QL-Ci4, the Hervey Circle.
  • QL-P3 is the center of the QL-Qu2, the Kantor-Hervey Deltoid.
  • QL-P3 is the Gergonne-Steiner Point (QA-P3) as well as the Isogonal Center (QA-P4) as well as the Midray Homothetic Center (QA-P8) as well as the QA-DT-Orthocenter (QA-P12) from the Orthocenter Quadrangle in the Circumcenter Quadrangle H1.H2.H3.H4. These 4 QA-points concur. H1.H2.H3.H4 is concyclic.
  • d (QL-P3, QL-P2) = 3 * d (QL-P5 , QL-P12).
  • d (QL-P3, QL-L1) = d (QL-P2, QL-L1) = d (QL-P4, QL-L1), where d=distance and QL-L1=Newton Line.
  • QL-P3 is the endpoint of the vector being the sum of the vectors from QL-P4 to the 4 Circumcenters of the QL-Component Triangles. See Ref-49.
  • The QL-P3-Triple Triangle in a Quadrangle is perspective with the QL-P2-Triple Triangle as well as the QL-P29-Triple Triangle with perspector QA-P15.

 
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