QL-12P1: A dozen of Equidistance Midpoints

 
As described by lines QL-12L1 (Equidistance Lines Dozen) there are 12 lines per quadrilateral such that each line is separated by the 4 lines in 3 equal parts.
It is interesting that these 3 parts have a midpoint. It is in fact the midpoint of the part in the middle.
Since there are 4 equidistance lines per quadrigon of the Reference Quadrilateral there are also 4 Equidistance Midpoints per Quadrigon of the Reference Quadrilateral.
The 4 Equidistance Lines per Quadrigon are those lines where the middle segment of the Equidistance Line is the segment between the opposite lines.
 
 
QL-12P1-EquiDistancePoints-01.fig
 
 
Coordinates:
The coordinates of the EquiDistance Midpoints are described in the paragraph of the EquiDistance Lines (QL-12L1).
 
 
Properties:
  • The 4 Equidistance Midpoints in a QL-Quadrigon lie on the corresponding Nine-point Conic (see QA-Co1) of the Quadrigon in question.
  • The 4 Equidistance Midpoints in a QL-Quadrigon form a trapezoid, where
m3241.m4132 // m2134.m1243 // 3rd diagonal of the Quadrigon.
  • m3241.m2134 ^ m4132.m1243 = QG-P2 (midpoint 3rd diagonal of the Quadrigon).
  • Let L1, L2, L3, L4 be the lines of a Quadrigon where L1, L3 are opposite sides and L2, L4 are opposite sides. Let P be a point on the Nine-point Conic of the Quadrigon and let L be a line through P. When P is the midpoint of the line segment of L between L1 and L3 then P is automatically the midpoint of the line segment of L between L2 and L4.
  • The Centroid of the 4 Equidistance Midpoints in a QL-Quadrigon lies on the Newton Line (see QL-L1).
  • The Centroid of the 12 Equidistance Midpoints of all 3 QL-Quadrigons is a point also on the Newton Line (QL-L1).
  • The Involutary Conjugate (QA-Tf2) of an Equidistance Midpoint (performed in the Quadrigon where the Equidistance Line is constructed) is the Infinity Point of the Equidistance Line (see QL-12L1) it is on.

 

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