QG-Co1: Inscribed Harmonic Conic

There is a projective transformation from 4 lines in a square to any other set of 4 lines.

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

The Inscribed Harmonic Conic is the projective transformation of the inscribed circle in a square to the Reference Quadrigon. This conic touches the sidelines of the Quadrigon in their perspective midpoints.

See picture below.

Construction QG-Co1

Equation QG-Co1 in 3 QA-Quadrigons in CT-notation:

q² r² x² + p² r² y² + p² q² z² – 6 p q r² x y + 2 p² q r y z + 2 p q² r x z

q² r² x² + p² r² y² + p² q² z² + 2 p q r² x y – 6 p² q r y z + 2 p q² r x z

q² r² x² + p² r² y² + p² q² z² + 2 p q r² x y + 2 p² q r y z – 6 p q² r x z

Equation QG-Co1 in 3 QL-Quadrigons in CT-notation:

l² x² + m² y² + 4 n² z² – 2 l m x y + 4 m n y z + 4 l n x z

4 l² x² + m² y² + n² z² + 4 l m x y – 2 m n y z + 4 l n x z

l² x² + 4 m² y² + n² z² + 4 l m x y + 4 m n y z – 2 l n x z

Equation QG-Co1 in 3 QA-Quadrigons in DT-notation:

q² r² x² – p² r² y² + p² q² z²

-r² q² x² + r² p² y² + p² q² z²

q² r² x² + r² p² y² – q² p² z²

Equation QG-Co1 in 3 QL-Quadrigons in DT-notation:

2 x² l² – y² m² + 2 z² n²

-x² l² + 2 y² m² + 2 z² n²

2 x² l² + 2 y² m² – z² n²

Properties

• The Center of QG-Co1 is QG-P12.
• The Diagonal Triangles QA-Tr1 and QL-Tr1 are self-polar wrt QG-Co1.
• The Triangle QG-P1.QG-P2.QG-P3 is self-polar wrt QG-Co1.

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