QA-Tf3: QA-Orthopole Transformation

Let P be a random point.
Let O1, O2, O3, O4 be the circumcenters of the Component Triangles of Reference Quadrangle P1.P2.P3.P4.
Let OPi be the Orthopole of line P.Oi wrt Component Triangle Pj.Pk.Pl, where (i,j,k,l) ∈ (1,2,3,4).
The 4 Orthopoles OP1, OP2, OP3, OP4 are concyclic.
For an explanation of Orthopole see Ref-13.
The circle through OP1, OP2, OP3, OP4 passes through QA-P2.
This construction was found by Antreas P. Hatzipolakis. See Ref-11, Hyacinthos messages # 21865 & 21867.
QA-Tf3 is the Center of the circle through OP1, OP2, OP3, OP4.
The coordinates of the transformed point show that QA-Tf3 is a linear transformation. When P = QA-P3   then the QA-Orthopole-circle = circle with diameter QA-P2.QA-P3.
When P = QA-P4   then the QA-Orthopole-circle = point QA-P2.
When P = QA-P12 then the QA-Orthopole-circle = circumcircle of the Diagonal Triangle.

There is another construction for QA-Tf3: And here is a special property of QA-Tf3: This transformation was earlier described by Benedetto Scimemi (called ‘p’) as a negative similarity transforming the Reference Quadrangle into the Quadrangle of the pedal centers. He refers to H.V. Mallison (Ref-46) who made notice of this transformation without many properties. Benedetto noted many more properties with proofs at Ref-36, pages 348-350, among of them (some not mentioned above):
• p is a negative similarity
• p transforms the reference quadrangle into the quadrangle of the pedal centers
• p transforms the quadrangle of the circumcenters into the quadrangle of the ninepoint centers, and therefore
p transforms QA-P3 into QA-P1
p transforms QA-P4 into QA-P2
p transforms QA-P8 into QA-P7
• p interchanges the asymptotic points of QA-Co2
• there is a fixed point of p: P
• p is the commutative product of a reflection on a line through P and a homothety centered in P
• the homothety coefficient equals 2 cos α , where α is the angle formed by the asymptotes of QA-Co1 (this holds for convex quadrangles; a similar formula holds for the non-convex case).
• Construction: Given any quadrangle A1A2A3A4 and a generic point Z. Reflect Z in two opposite sides of the quadrangle and take the midpoint, say M12,34=midpoint (ZA1A2, ZA3A4). Define similarly M13,24and M14,23Then p(Z) is the circumcenter of the triangle M12,34M13,24M14,23.

Transformation:
1st coordinate QA-Tf3 in CT-notation, given point (u:v:w) to be transformed:
a2 (p r v + q r v + p q w + q r w) + b2 p (r u + r v - q w) + c2 p (q u + q w - r v)
1st coordinate QA-Tf3 in DT-notation, given point (u:v:w) to be transformed:
a2 (-p2 u - p2 v + r2 v - p2 w + q2 w) + b2 p2 (u + v) + c2 p2 (u + w)
Examples of QA-Orthopoles:
The following table lists QA-Orthopoles of several QA-/QG-/QL-points.

 QA-Point QA-Orthopole QA-P1: Quadrangle Centroid Midpoint (QA-P1, QA-P23) QA-P2: Euler-Poncelet Point QA-P23: Inscribed Square Axes Crosspoint QA-P3: Gergonne-Steiner Point QA-P1: Centroid QA-P4: Isogonal Center QA-P2: Euler-Poncelet Point QA-P6: Parabola Axes Crosspoint Midpoint (QA-P2, QA-P23) QA-P7: QA-Nine-point Center Hom.Center Point on Line QA-P8.QA-P23  ( 2 : 1) QA-P8: Midray Homothetic Center QA-P7: QA-Nine-point Center Hom.Center QA-P12: Orthocenter QA-Diag. Triangle QA-P11: Circumcenter QA-Diag. Triangle QA-P32: Centroid of Circumcenter Quadr. QA-P33: Centroid of Orthocenter Quadr. Point on Line QA-P23.QA-P33 (-1 : 3) QA-P36: Complement QA-P30 wrt QA-DT Midpoint (QA-P11, QA-P23) QA-P42: QA-Orthopole Center QA-P42: QA-Orthopole Center QG-Point QG-P1: Diagonal Crosspoint Point on Newton Line (QL-L1) QG-P5: 1st QG-Quasi Circumcenter QG-P7: 1st QG-Quasi Nine-point Center QG-P17: Projection QG-P1 on QG-L1 Midpoint (QG-P1, QG-P18) QL-Point QL-P1: Miquel Point Point on Newton Line (QL-L1)

The following table lists interesting QA-Orthopoles of several QG-/QL-Lines & Curves.
 QG-Line QA-Orthopole Line QG-P1.QL-P1 QL-L1: Newton Line QL-Circles QL-Ci5: Plücker Circle Circle through QL-P5 (Center of QL-Ci5) QL-Ci6: Dimidium Circle Circle through QA-P1

Other Properties:
• The reflection axis of QA-Tf3 and its perpendicular are parallel to:
- the axes of the QA-Nine-point Conic (QA-Co1)
- the asymptotes of the QA-Orthogonal Hyperbola (QA-Co2)
- the axes of the Gergonne-Steiner Conic (QA-Co3)
• The QA-Tf3 image of a point on the internal or external angle bisector of QA-P2.QA-P42.QA-P4 at QA-P42 lies on that same angle bisector.
• Let A1.A2.A3.A4.A5 . . . . be a random polygon with vertices Ai (i = 1,2,3,4,5, . . .).  Let Ti be the QA-Orthopoles of Ai (i = 1,2,3,4,5, . . . . ). Now polygon A1.A2.A3.A4. . . . is similar to polygon T1.T2.T3.T4. . . . , reflected in the internal angle bisector of QA-P2.QA-P42.QA-P4 at QA-P42.
• The circle described in the construction of QA-Tf3(P) is also the locus of QA-Tf6 wrt a pencil of lines through random point P.
• Reflect some random point P in two opposite sides of the quadrangle and let M12,34 be their midpoint of these reflections. Define similarly M13,24 and M14,23. Then QA-Tf3(P) is the circumcenter of the triangle M12,34M13,24M14,23 and its circumcircle coincides with the circle described in the initial construction of QA-Tf3.
• The circle described in the construction of QA-Tf3(P) is also used in the construction of 5P-s-Tf3.
• QA-Tf6(Lp) will be a point on the circle described in the construction of QA-Tf3(P) for any line Lp through P. See Ref-34, Ngo Quang Duong, message #2745.

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