QL-L3: QL-Pedal Line

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The perpendicular feet from the Miquel Point (QL-P1) to the four lines lie on the same line R, and the Miquel Point is the only point with this property.

This is statement 3 of Steiner’s note on the complete quadrilateral as described on page 1 of Ref-4. This line R is now according to Ref-4 named the Pedal Line.

QL-L3 is also called the “pedal line” by J.W. Clawson in Ref-31.

Coefficients: 

1st CT-coefficient:

            l (m - n) / (a² m n - S_B l m - S_C l n)

1st DT-coefficient:

            (m²-n²)³ SA² + (l²-n²)³ SB² + (m²-l²)³ SC² + 2 (m²-n²) SA ((l²-n²)² SB+(l²-m²)² SC)

Properties:

• QL-P19 lies on QL-L3.
• QL-L3 is parallel to QL-L2 (Steiner Line).
• QL-L3 is perpendicular to QL-L1 (Newton Line) and QL-L4 (Morley Line).
• QL-L3 is tangent to the QL-Inscribed Parabola QL-Co1 at the vertex.
• QL-Co1 is the 5^th Line Conic (see QL-Co-1) of QL-L3.
• The Orthopole of ANY line through the Miquel Point (QL-P1) wrt ANY Component Triangle of the Reference Quadrilateral lies on QL-L3. See Ref-33, Anopolis # 637.
• The Orthopole of ANY line parallel to the Newton line (QL-L1) wrt ANY Component Triangle of the Reference Quadrilateral lies on QL-L3. See Ref-34, Seiichi Kirikami, QFG message # 1102.
• The Orthopoles of QL-L3 wrt the QL-Component Triangles lie on QL-L2. See Ref-34, Seiichi Kirikami, QFG message # 1102.
• QL-L3 is the common Simson Line of QL-P1 wrt the 4 QL-Component Triangles. See Ref-34, QFG messages #551, #1105.
• The line through the intersection point of Li and Lj (i,j=1,2,3,4) and through the intersection point of the reflected lines of QL-L3 in Li and Lj passes through QL-P1. See Ref-34, Seiichi Kirikami, QFG message # 1096.

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