QL-Ci6: Dimidium Circle


The Dimidium Circle is the circle through the Gergonne-Steiner Points (QA-P3) of the 3 component Quadrigons of the Reference Quadrilateral.  Its center is QL-P6.
The word “Dimidium” is the Latin word for "half".
Somehow of all Quadrilateral Circles this circle has the simplest algebraic CT-equation.
 
QL-Ci6-DimidiumCircle-00

Equations: 
Equation in CT-notation:
     l  (m - n) (b2 (l - m) + c2 ( l  -  n)) x2   -  (m - n) (3 a2 (l  - m) (l  - n) + b2 n (l - m) + c2 m (l - n)) y z
+ m (n  -  l) (c2 (m - n) + a2 (m -  l)) y -  (n -  l ) (3 b2 (m - l) (m - n) + c2 l (m - n) + a2 n (m - l)) z x
+ n  (l  - m) (a2 (n -  l ) + b2 (n - m)) z2   -  ( l - m) (3 c2 (n - m) (n -  l) + a2 m (n - l) + b2 l (n - m)) x y   =   0
Equation in DT-notation:
   (m2 - n2) (b2 ( l2 - m2) + c2 ( l2 - n2))  l2  x2  +  (m2 - n2) (-a2  l4  + (a2 - c2)  l2 m2 +  (a2 - b2)  l2  n2  + 2 SA m2 n2) y z
+ (n2 -  l2 ) (c2 (m2 - n2) + a2 (m2 - l2)) m2 y2  +  (n2 -  l2 ) (-b2 m4 + (b2 - a2) m2 n2 + (b2 - c2)  l2 m2 + 2 SB  l2   n2) x z
+ (l2 - m2) (a2 (n2 -  l2 ) + b2 (n2 - m2)) n2 z2   +  (l2 - m2)  (-c2 n4  + (c2 - b2)  l2  n2 +  (c2 - a2) m2 n2 + 2 SC  l2  m2) x y  =  0

Properties:
  • QL-Ci6 passes through QL-P1( the Miquel Point) as well as the 3 QL-versions of QA-P3.
  • QL-Ci6 passes also through QL-P17 (QL-Adjunct Quasi Circumcenter) and QL-P24 (Intersection QL-P1.QL-P8 ^ QL-P13.QL-P17) which are the intersection points of QL-Ci6 with QL-Ci1 (Circumcircle of the QL-Diagonal Triangle).
  • The Clawson-Schmidt Conjugate of QL-P26 lies on QL-Ci6. It is the 2nd intersection point of QL-P1.QL-P13 with circle QL-Ci6.
  • The center of QL-Ci6 is QL-P6 (the Dimidium Point). This 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: S1, S2, S3. These points form triangle QL-Tr2 and lie on the Dimidium Circle.
  • Every QL-Component Triangle has a conic Coi (i=1,2,3,4) through its vertices, centroid and its perspector with the QL-Diagonal Triangle. These 4 conics Coi (i=1,2,3,4) also have 3 common points lying on QL-Ci6, being the vertices of QL-Tr2. See Ref-34, Bernard Keizer, QFG-messages #457, #1458, also for more properties.
  • Per QL-Component Triangle CTi (i=1,2,3,4) the 2nd intersectionpoint of the CTi-circumcircle with the line through QL-P1 and the perspector of CTi with the QL-Diagonal Triangle is resident on the Dimidium Circle. See Ref-34, QFG-messages #457, #1458, #1466 of Bernard Keizer.
  • The Dimidium Circle (QL-Ci6) lies exactly between the Plücker Circle (QL-Ci5) and the Miquel Circle (QL-Ci3). This is a set of 3 coaxal circles. One of their common points is QL-P1 (Miquel Point).
  • The 3 QA-versions of QL-Ci6 meet in QA-P3 (Gergonne-Steiner Point) (note Eckart Schmidt).
  • When one of the 3 Component QL-Quadrigons is supposed to be a QA-Quadrigon, the midpoints of the Miquel Points (QL-P1) and the corresponding Diagonal Crosspoints (QG-P1) in the other 2 QA-Quadrigons also lie on the Dimidium Circle.
  • The three QG-P16 points in a quadrilateral are collinear on a line through QL-P26, which is the QL-Tf1 image of the Dimidium circle QL-Ci6 (Eckart Schmidt, November 26, 2012).
  • The QA-Orthopole (QA-Tf3) of QL-Ci6 is a circle through QA-P1.
  • The 2nd intersection point of QL-Ci2 and QL-Ci6 lies on QL-P8.QL-P17.QL-P25. See Ref-34, Eckart Schmidt, QFG-message #1666.

 

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