QG-Co2: Circumscribed Harmonic Conic

The Circumscribed Harmonic Conic is the projective transformation of the circumscribed circle of a square to the Reference Quadrigon.

See chapter QG-Tf1: QG-Projective Square Transformation.

This conic touches the sidelines of the projective circumscribed Quadrigon at the vertices of the Reference Quadrigon.

See picture below.

Construction QG-Co2

Equation QG-Co2 in 3 QA-Quadrigons in CT-notation:

-2 r x y + q x z + p y z

r x y – 2 q x z + p y z

r x y + q x z – 2 p y z

Equation QG-Co2 in 3 QL-Quadrigons in CT-notation:

l² x² + l m x y + l n x z – m n y z

m² y² + l m x y – l n x z + m n y z

n² z² – l m x y + l n x z + m n y z

Equation QG-Co2 in 3 QA-Quadrigons in DT-notation:

q² r² x² – 2 p² r² y² + p² q² z²

-2 q² r² x² + p² r² y² + p² q² z² 

q² r² x² + p² r² y² – 2 p² q² z²

Equation QG-Co2 in 3 QL-Quadrigons in DT-notation:

x² l² – y² m² + z² n²

-x² l² + y² m² + z² n² 

x² l² + y² m² – z² n² = 0

Properties

• The Center of QG-Co2 is QG-P13.
• The Diagonal Triangles QA-Tr1 and QL-Tr1 are self-polar wrt QG-Co2 (Eckart Schmidt, August 24, 2012).
• The Triangle QG-P1.QG-P2.QG-P3 is self-polar wrt QG-Co2.
• QL-P1 lies on the polar of QG-P16 (and invers) wrt QG-Co2. See [34] Eckart Schmidt , October 9, 2013, QFG-message # 286.
• The polar of QG-P12 wrt QG-Co2 is parallel to QG-L1 (QG-P1-Railway Watcher, see QL-L-1). See [34] Eckart Schmidt , October 9, 2013, QFG-message # 286.
• The polar of QG-P3 wrt QG-Co2 is QG-P1.QG-P2. See [34] Eckart Schmidt , October 9, 2013, QFG-message # 286.

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