QA-Tf6 Quang Duong’s Transformation

This Transformation was introduced by
Ngo Quang Duong. See Ref-34, QFG#1176.
Let ABCD be the Reference Quadrangle and r a random line.
Line r intersects AB, CD, AC, DB, AD, BC at MAB, MCD, MAC, MDB, MAD, MBC.
HAB, HCD, HAC, HDB, HAD, HBC is orthogonal projections of MCD, MAB, MDB, MAC, MBC, MAD on  AB, CD, AC, DB, AD, BC.
Then HAB HCD, HAC HDB, HAD HBC are concurrent in a point R being the Quang Duong’s Transformation of line r wrt Reference Quadrangle ABCD.
When vertices A,B,C,D are concyclic, then the lines HAB HCD, HAC HDB, HAD HBC will be parallel and consequently R will be an infinity point. Alternative constructions (see Ref-34, Benedetto Scimemi, QFG # 1185):
1. QA-Tf6(r) = QA-Tf3 (reflection of QA-P4 in r)
2. QA-Tf6(r) = reflection of QA-P2 in QA-Tf3(r)

Tables:
Following tables list the relations between several Quadrangle Objects.
General Behavior of the transformation. See Ref-34, ES#1187,1191.
 Item Transformed Item Pencil of parallel lines Line through QA-P2 Pencil of lines perpendicular to some line L1 L2=Line Pencil of lines perpendicular to line L2 Line through QA-P2 parallel to L1 Line pencil through some point P Ci=Circle through QA-P2 Line pencil through QA-Tf6(P) perpendicular to r Envelope=Deltoid circumscribing Ci Line pencil through QA-Tf6(P) parallel to r Envelope=Deltoid circumscribing Ci, homothetic with the deltoid for the Simson lies of the Diagonal Triangle (wrt QA-P12 and factor 2)

Transformation of classes of QA-objects. See Ref-34, ES#1178,1191
 Item Transformed Item Line through QA-P4 Line pencil through QA-P1 Circle through QA-P2 centered in midpoint QA-P1.QA-P23 Line pencil through QA-P2 Circle through QA-P2 centered in QA-P23 Line pencil through QA-P12 Circumcircle of Diagonal triangle (QA-Ci1) Pencil of lines perpendicular to QA-L4 Pencil of lines perpendicular to QA-L1 Line through QA-P2 parallel to QA-L4 Pencil of lines perpendicular to QA-L2 Pencil of lines perpendicular to QA-P2.QA-P23 Tangents to circles (Ci1) round QA-P4 Circles (Ci2) round QA-P2 Intersection of circles Ci1 and Ci2 Circle through QA-P3 centered in QA-P2.QA-P4 ^ QA-P3.QA-P32 Radical Axes of Ci1 and Ci2 Tangents to Parabolas with focus QA-P4 Perpendicular bisector of QA-P2 and the QA-Tf6-image of the parabola directrix Tangents to a Conic with focus QA-P4 Circle centered in the QA-Tf6-image of the secondary axis of the conic

Transformation of specific QA-Objects. See Ref-34, #1181(CvT), #1191(ES).
 Item Transformed Item QA-L1 (line through QA-P1,QA-P2,QA-P3) Point on QA-P3.QA-P4 QA-L2 (line through QA-P2,QA-P4,QA-P6) QA-L4 (line through QA-P1,QA-P6,QA-P23) Point on line through QA-P23 parallel to QA-L1 QA-L5 (Eulerline of Diagonal Triangle) Point on circumcircle Diagonal Triangle QA-L9 (perpendicular bisector QA-P2.QA-P4) Line through QA-P3 perpendicular to QA-L4

1st CT-coordinate for line (x:y:z):
- a2 (a2 p q - b2 p q + c2 p q + a2 p r + b2 p r - c2 p r + 2 a2 q r) x2
- b2 (a2 - b2 + c2) p q y2
- c2 (a2 + b2 - c2) p  r z2
- (a2 c2 q + b2 c2 q - c4 q + a2 b2 r - b4 r + b2 c2 r) p y z
+ (a4 p q - b4 p q + 2 b2 c2 p q - c4 p q + 2 a2 b2 p r + a4 q r + a2 b2 q r - a2 c2 q r) x y
+ (2 a2 c2 p q + a4 p r - b4 p r + 2 b2 c2 p r - c4 p r + a4 q r - a2 b2 q r + a2 c2 q r) x z

Properties:
• QA-Tf6(d) is a point on the QA-Orthopolar Line QA-Tf8(d). See Ref-34, QFG#1186). For definition Orthopolar Line see Ref-13.
• 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. QA-Tf6(Lp) will be a point on this circle for any line Lp through P. See Ref-34, Ngo Quang Duong, message #2745.

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