QL-L4: Morley Line

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The Morley Line is the line through QL-P2 (Morley Point) and QL-P3 (Kantor-Hervey Point).

 

Coefficients: 

1st CT-coefficient:

+a⁴ (l - m) (l - n) (m - n) (l m + l n - m n)

+b⁴ (l - m) (l - n) (m - n) l m

+c⁴ (l - m) (l - n) (m - n) l n

+b² c² (n – m) U0

+a² c² ( l – n) (U0 – U1)

+a² b² (m – l) (U0 – U2)

where:

U0 = l (l² m + l² n - 4 l m n + m² n + m n²)

U1 = (l - m) (l + m - 2 n) (l m + l n - m n)

U2 = (l – n) (l + n - 2 m) (l m + l n - m n)

1st DT-coefficient:

            l² (m²-n²)³ SA² - m² (l²-n²)³ SB² + n² (l²-m²)³ SC²

   + (l²-m²) (l²-n²) (m²-n²) (2 l² S² - l² SB SC - m² SC SA- n² SA SB)

Properties:

• QL-L4 is parallel to QL-L1 (Newton Line).
• QL-L4 is perpendicular to QL-L2 (Steiner Line) and QL-L3 (QL-Pedal Line).
• QL-L4 is parallel to the Trilinear Polar of QL-P24 (note Eckart Schmidt).
• The 3 QL-versions of QA-P15 lie on QL-L4 (Eckart Schmidt, August 24, 2012).

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