QA-L6: QA-Newton-Morley Line

Each QL-line parallel to the Newton Line when transferred to the 3 Quadrigons of a Quadrangle produces 3 concurrent lines. The locus of the common point is QA-L6.

QA-P1 (the Quadrangle Centroid) is a point on this line because the QL-Newton Line is transferred into QA-P1.

1st CT-coefficient:

-a⁴ q r (p + q) (q – r) (p + r)

-c⁴ (p + r) (q + r) (p³ + 3 p² q + 4 p q² + 2 q³ + 2 p² r + 3 p q r + 3 q² r + p r² + q r²)

+b⁴ (p + q) (q + r) (p³ + 2 p² q + p q² + 3 p² r + 3 p q r + q² r + 4 p r² + 3 q r² + 2 r³)

+b² c² (q – r) (q + r) (-p² q – p q² – p² r + q² r – p r² + q r²)

+a² c² (p + r) (p³ q + 3 p² q² + 4 p q³ + 2 q⁴ + p³ r + 7 p² q r + 12 p q² r + 6 q³ r + 2 p² r² + 7 p q r² + 7 q² r² + p r³ + 3 q r³)

-a² b² (p + q) (p³ q + 2 p² q² + p q³ + p³ r + 7 p² q r + 7 p q² r + 3 q³ r + 3 p² r² + 12 p q r² + 7 q² r² + 4 p r³ + 6 q r³ + 2 r⁴)

1st DT-coefficient:

(-c² q²+b² r²) (p² (p⁴-3 (q²-r²)²+2 p² (q²+r²)) SA+(3 p⁴-(q²-r²)²-2 p² (q²+r²)) (q² SB+r² SC))

Properties

• Since the QL-Morley Line is parallel to the QL-Newton Line also the common QA-point of the Morley lines QA-P15 lies on QA-L6.

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