**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.

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.

*Coordinates:**1st CT-coordinate:*

a

^{2}(l - m) (l - n) (b^{2}n - c^{2}m) + (b^{2}- c^{2}) (m - n) (-b^{2}(l - m) n + c^{2}m (l - n))*1st DT-coordinate:*

a

^{4}- b^{4}(3 m^{2}+n^{2})/(l^{2}-n^{2}) - c^{4}(m^{2}+3 n^{2})/(l^{2}-m^{2})- 4 b

^{2}c^{2}m^{2}/(-l^{2}+m^{2}) - 4 b^{2}c^{2}n^{2}/(-l^{2}+n^{2}) + 2 a^{2}(b^{2}-c^{2}) (m^{2}+n^{2})/(m^{2}-n^{2})

*Properties:*-
QL-P3 lies on this line:

- 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.