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.
The coordinates of the EquiDistance Midpoints are described in the paragraph of the EquiDistance Lines (QL-12L1).
  • 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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