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QL-L2: Steiner Line

The Orthocenters of the 4 component triangles of a Quadrilateral are collinear. The line through these 4 Orthocenters is the Steiner Line.

The Steiner Line is also known as Ortholine.

QL-L2 is also called the Aubert Line. See [72], Euclid-messages #2420, #2435 and [66], QPG-messages #1149-#1151.

QL-L2 is also called the “orthocentric line” by J.W. Clawson in [31].

Jean-Louis Ayme wrote [2a] about the Newton Line as well as the Steiner Line.

Infovisual QL-L2-infovisual-cvt-01.png
1st CT-coefficient

SA l (m – n)

1st DT-coefficient

b2 (l2-m2) – c2 (l2-n2)

Properties
  • QL-P2 (Morley Point) and QL-2P1a/b (1st/2nd Plücker Point) and QL-P7 (Newton-Steiner Point) and QL-P9 (Circumcenter QL-Diagonal Triangle) lie on QL-L2.
  • QL-L2 is parallel to QL-L3 (QL-Pedal Line).
  • QL-L2 is perpendicular to QL-L1 (Newton Line) and QL-L4 (Morley Line).
  • QL-L2 is the directrix of QL-Co1, the Inscribed Quadrilateral Parabola.
  • QL-L2 is the common radical axis of the three circles constructed on the diagonals of the Reference Quadrilateral as diameters. This is also known as the Gauss-Bodenmiller theorem (see [13]).
  • QL-L2 is the Clawson-Schmidt Conjugate (QL-Tf1) of QL-Ci3 (Miquel Circle).
  • The Orthopole of a sideline of the complete quadrilateral with respect to the triangle bounded by the three other sidelines lies on the Steiner line (see [4] page 42).
  • The Orthopole of a line L wrt a Triangle = the common intersection point of the three lines perpendicular to the sidelines of the triangle, each passing through the projection of the opposite vertex on line L.
  • The Orthopoles of QL-L3 (Pedal Line) wrt the QL-Component Triangles lie on QL-L2. See [34], Seiichi Kirikami, QFG message # 1102.
  • The reflections of QL-P1 in the 4 basic lines of the Reference Quadrilateral lie on QL-L2. See [34], Seiichi Kirikami, QFG message # 1091.
  • The reflections of QL-L2 in the 4 basic lines of the Reference Quadrilateral concur in QL-P1. See [34], Seiichi Kirikami, QFG message # 1095.
  • QL-L2 is the common Steiner Line (in a triangle) of QL-P1 wrt the 4 QL-Component Triangles (Def.: The reflections of a point P of the circumcircle on the sides of the triangle lie on a line called the Steiner line of P wrt that triangle). See [34], QFG #1105.


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