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).

1^st 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 m (m – n) (n – l) + a² b² (l – m) (- l n + 3 l m – 6 m n+ 4 n²)

1^st 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 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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