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QG-P3 Midpoint 3rd QL-Diagonal

We use this terminology.

  • A “QA-Quadrigon” is a Quadrigon seen as the Component of a Quadrangle.
  • A “QL-Quadrigon” is a Quadrigon seen as the Component of a Quadrilateral.

The 3rd Diagonal of a QA-Quadrigon is the same line as the 3rd Diagonal of a QL-Quadrigon. However the Midpoint of a QA-Quadrigon is different from the Midpoint of a QL-Quadrigon.

QG-P3 is the Midpoint of the segment on the 3rd Diagonal of a QL-Quadrigon limited by the intersection points with the 3rd Diagonals of the 2 other Component QL-Quadrigons.

Construction

Let S1 = P1.P2 ^ P3.P4 and S2 = P2.P3 ^ P4.P1. S1.S2 is the 3rd Diagonal.

Let S3 = P1.P3 ^ S1.S2 and S4 = P2.P4 ^ S1.S2.

QG-P3 = the Midpoint of S3 and S4.

Infovisual QG-P3-infovisual-cvt-01.png
CT-Coordinates QG-P3 in 3 QA-Quadrigons

( p (p + r) : -q (q + r) : r (p – q) )

(-p (p + q) : q (-p + r) : r (q + r) )

( p (q – r) : q (p + q) : -r (p + r) )

CT-Coordinates QG-P3 in 3 QL-Quadrigons

(m2 n2 : -n l2 (n – m) : m l2 (n – m))

(m2 (l – n) n : l2 n2 : -l m2 (l – n))

(-m n2 (m – l) : l (m – l) n2 : l2 m2 )

CT-Area of QG-P3-Triangle in the QA-environment: (points are collinear)

0

CT-Area of QG-P3-Triangle in the QL-environment: (equals ¼ x area QL-Diagonal Triangle)

l2 m2 n2 / ((-l m + l n + m n) (l m + l n – m n) (l m – l n + m n))

–

DT-Coordinates QG-P3 in 3 QA-Quadrigons

(0 : -q2 : r2)

(-p2 : 0 : r2)

(-p2 : q2 : 0)

DT-Coordinates QG-P3 in 3 QL-Quadrigons

(0 : 1 : 1)

(1 : 0 : 1 )

(1 : 1 : 0)

DT-Area of QG-P3-Triangle in the QA-environment:(points are collinear)

0

DT-Area of QG-P3-Triangle in the QL-environment:(equals ¼ x area QL-Diagonal Triangle)

S / 8

Properties
  • QG-P3 lies on these lines:
  • The Triangle formed by the 3 QL-Versions of QG-P3 is the medial triangle of the QL-Diagonal Triangle.
  • The 3 QA-Versions of QG-P3 are collinear points.
  • The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1. The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1. The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1 The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1 The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1 The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1The three QA-versions of QG-P3 are collinear on the tripolar of QA-P16 wrt QA-Tr1. See [34], Eckart Schmidt, QFG#1263.
  • The Polar (see [13], Polar) of QG-P3 wrt QG-Co1 as well as QG-Co2 is the line QG-P1.QG-P2.


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