QL-P6: Dimidium Point


The Dimidium Point is the Circumcenter of the Dimidium Circle.
I call this point the Dimidium Point because “Dimidium” is the Latin word for "half".
This is because:
  • The Dimidium Point is the midpoint of QL-P4 (Miquel Circumcenter) and QL-P5 (Clawson Center).
  • The intersection points of the Nine-point Conics of the 3 component quadrigons of the Reference Quadrilateral have 3 common points: Sn1, Sn2, Sn3. These points lie on the Dimidium Circle (which moreover passes through QL-P1 the Miquel Point). As is known the Conic is a conic through all midpoints of all line segments in a quadrangle or quadrigon.
  • The Gergonne-Steiner Points Mia, Mib, Mic of the 3 component quadrigons of the Reference Quadrilateral also lie on the Dimidium Circle. As is known the Gergonne-Steiner Point (QA-P3) is constructed from the midpoints of a pencil of 3 line segments.
  • The Dimidium Circle (QL-Ci6) lies exactly between (midway) the Plücker Circle (QL-Ci5) and the Miquel Circle (QL-Ci3).

QL-P6-DimidiumPoint-00
 
Coordinates: 
1st CT-coordinate:
                     3 a4 (l - m) (m - n) (n - l) + b2 c2 (m - n) (- l m - l n +2 m n)
   +   b4 (l - m) (m - n)   n        + c2 a2 (n - l) (- l m + 3 l n - 6 m n+ 4 m2)
   +   c4  (l - n) (m - n)   m       + a2 b2 (l - m) (- l n + 3 l m - 6 m n+ 4 n2)
1st DT-coordinate:
            Sb2 (l2-n2)2 m2 - Sc2 (l2-m2)2 n2
+ (m2-n2) (Sb Sc l4 +Sa Sb l2 n2+Sa Sc l2 m2 +S2(l2 m2+l2 n2-3 m2 n2))

Properties:
  • QL-P6 lies on these lines:
        QL-P2.QL-P12                 (3 : 1)
        QL-P3.QL-P4                   (3 : 1)
  • QL-P6 = Midpoint of QL-P4 and QL-P5.
  • Distances QL-P6.QL-P12 : QL-P12.QL-P2 = 1 : 2.
  • QL-P6 = Center of QL-Ci6, the Dimidium Circle.
  • QL-P6 = Circumcenter of QL-Tr2.
  • QL-P6 = Circumcenter of the 3 QL-versions of QA-P3 (Eckart Schmidt, July, 2012).
  • QL-P6 = Centroid of the Triangle formed by the 3 QL-versions of QA-P32 (note Eckart Schmidt).
  • QL-P6 = the Quadrangle Centroid (QA-P1) from the Circumcenter Quadrangle (O1.O2.O3.O4 in picture QL-P3).
  • QL-P6 = the perspector of the Centroid Quadrangle of the Circumcenter Quadrangle of the Reference Quadrilateral and the Circumcenter Quadrangle of the Reference Quadrilateral (note Seiichi Kirikami).
     

 

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