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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.

Infovisual QL-Ci6-infovisual-cvt-01.png
Equation in CT-notation

l (m – n) (b2 (l – m) + c2 (l – n)) x2

+ m (n – l) (c2 (m – n) + a2 (m – l)) y2

+ n (l – m) (a2 (n – l) + b2 (n – m)) z2

– (m – n) (3 a2 (l – m) (l – n) + b2 n (l – m) + c2 m (l – n)) y z

– (n – l) (3 b2 (m – l) (m – n) + c2 l (m – n) + a2 n (m – l)) z x

– (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 the 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 [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 [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) of 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 [34], Eckart Schmidt, QFG-message #1666.


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