QA-P23: Inscribed Square Axes Crosspoint

 
Each quadrigon has 2 Inscribed Squares. The centers of these squares (QG-2P1a and QG-2P1b) are connected by the so-called Inscribed Square Axis (QG-L6).
The Inscribed Square Axes of the 3 QA-Quadrigons of the Reference Quadrangle concur in one point. This point is QA-P23. It has very simple coordinates.
 
 
 QA-P23-InscrSquareAxesCrosspoint-01             
Coordinates:  
1st CT-Coordinate:
            a2 q r + SB p q + SC p r
1st DT-Coordinate:
            p2 SA
 
Properties:
  • QA-P23 lies on these QA-lines:
        QA-P1.QA-P6
        QA-P4.QA-P8
  • QA-P23 is concyclic with QA-P2, QA-P7 and QA-P8.
  • QA-P23 is the Involutary Conjugate (see QA-Tf2) of QA-P12.
  • QA-P23 is the QA-Orthopole (QA-Tf3) of QA-P2.
  • The Involutary Conjugate (QA-Tf2) of any line through QA-P23 is an Orthogonal Hyperbola (just like in a triangle the Isogonal Conjugate of any line through X(3) is an Orthogonal Hyperbola).
  • QA-P23 is the Orthology Center of the QG-P18 Triple Triangle wrt the QG-P17 Triple triangle. See QA-Tr-1.
  • QA-P23 is the Centroid of the Quadrangle of the Isogonal Conjugates (Benedetto Scimeni, October 2, 2015).
  • In any Quadrangle the circle centered in QA-P42 passing through QA-P2 inverts QA-P4 into QA-P23. (Benedetto Scimeni, October 2, 2015).

 
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