QA-P23: Inscribed Square Axes Crosspoint

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.
More information can be found at Ref-14 and Ref-29.

Coordinates:

1^st CT-Coordinate:

a² q r + S_B p q + S_C p r

1^st DT-Coordinate:

p² S_A

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).

MATHEMATICS