QA-Tf2: Involutary Conjugate

The Involutary Conjugate of a point Q is the 2nd Double Point of the QA-Line Involution (see QA-Tf1) occurring on the tangent at Q to the conic (P1, P2, P3, P4, Q).
It is a conjugate because applying two times this transformation ends up in the original point.
QA-Tf2 in a Quadrangle is the equivalent of QL-Tf2 in a Quadrilateral.
The Involutary Conjugate can also be seen as the Pi-Ceva Conjugate of Q wrt Component Triangle Pj.Pk.Pl (for all combinations of (i,j,k,l)  (1,2,3,4)) (note Bernard Gibert).
Construction 1
1. Construct the conic through P1, P2, P3, P4, Q.
2. Construct the tangent Lq at Q to this conic.
3. Construct the Involution Center IC of the created QA-Line Involution for Lq (see Ref-19 as well as QA-Tf5).
4. Now R = the Involutary Conjugate of Q = the Reflection of Q in IC.
Construction 2 (by Eckart Schmidt)
1. Choose one of the Component Quadrigons of the Reference Quadrangle.
2. Let Si and Sj be the intersection points of the opposite sides of the Quadrigon.
3. Connect X (the point to be transformed) with Si and Sj and construct on these lines the 4th harmonic points Xi and Xj wrt the intersection points with the crossing opposite quadrigon lines.
4. Connecting lines Si.Xj and Sj.Xi intersect in X*, the Involutary Conjugate of X.
Construction 3
1. Let S1 = P1.P2 ^ P3.P4, S2 = P1.P4 ^ P2.P3, P is a random point.
2. QA-Tf2(P) = 4th intersection point of conics (P1,P3,S1,S2,P) and (P2,P4,S1,S2,P).
(See Ref-34, Seiichi Kirikami, QFG-messages #1491, #1492)
Construction of the Involutary Conjugate of the Infinity Point of some line PQ
The involutary conjugate of the infinity point of some line Lpq through P and Q will lie on the involutary conjugate of the whole line, which is a QA-DT-conic through QA-Tf2(P) and QA-Tf2(Q), here called QA-Tf2(Lpq). Well known is that the locus of involutary conjugates of all infinity points is the nine-point conic QA-Co1. Therefore the involutary conjugate of the infinity point of some line P.Q is the 4th intersection point of QA-Co1 and QA-Cox. The other three intersection points are the vertices of the QA-Diagonal Triangle QA-DT.

QA Tf2 InvolutaryConjugate 40  of Infinity Point

Coordinates and Coefficients
Let Q (u : v : w) be a random point not on one of the connecting lines of the Reference Quadrangle.
It is possible to construct a conic through P1, P2, P3, P4, Q since a conic is defined by 5 points.
This tangent (see QA-L-1) at Q creates a QA-Line Involution (QA-Tf1), where Q represents the 1st Double Point and QA-Tf2 is the 2nd Double Point.
Let Q(u:v:w) be a random point.
Coordinates of QA-Tf2 in CT-notation:
  ( u (q r u - p r v - p q w) : v (-q r u + p r v - p q w) : w (-q r u - p r v + p q w) )
Coordinates of QA-Tf2 in in DT-notation:
                (p2 v w : q2 w u : r2 u v)
The following table lists a number of Involutary Conjugated pairs of points.                    
QA-P1: QA-Centroid
QA-P20: Reflection of QA-P5 in QA-P1
QA-P4: Isogonal Center
QA-P41: Involutary Conjugate of QA-P4
QA-P5: Isotomic Center
QA-P17: Involutary Conjugate of QA-P5
QA-P6: Parabola Axes Crosspoint
QA-P30: Reflection of QA-P2 in QA-P11
QA-P10: Centroid QA-Diagonal Triangle
QA-P16: QA-Harmonic Center
QA-P12: Orthocenter QA-Diag. Triangle
QA-P23: Inscribed Square Axes Crosspoint
QA-P18: Involutary Conjugate of QA-P19
QA-P19: AntiCompl. of QA-P16 wrt QA-DT
QA-P21: Reflection of QA-P16 in QA-P1
QA-P27: M3D Center
The Involutary Conjugate also can be applied to Quadrigon Points:
QG-P12: Inscribed Harmonic Conic Center
Reflection of QG-P1 in QG-P2
QG-P13: Circumscr.Harmonic Conic Center
Infinity Point of the line QG-P1.QG-P2
QG-P14: Center of the M3D Hyperbola
Infinity Point of the line QG-P1.QG-P3
QG-P15: Kirikami Center
Infinity Point of Newton Line QL-L1
QG-P16: Schmidt Point
QL-P1: Miquel Point
QG-P18: Quasi Isogonal Crosspoint
QG-P19: Quasi IsogonalConjugate of QG-P1
Next table lists a number of Involutary Conjugates of QA-lines and QA-curves.
The Involutary Conjugate transforms lines into circumscribed conics of the QA-Diagonal Triangle.
A “5th point tangent” (see QA-Tf9) at Q is transformed into a circumscribed conic of the QA-Diagonal Triangle through Q and its Involutary Conjugate.

Line at Infinity
QA-Co1: Ninepoint Conic
QA-L9: Perpendicular bisector QA-P2.QA-P4
QA-Ci1:  Circumcircle Diagonal Triangle
QA-L3: QA-Centroids Line
Circumscribed DT-Conic
through vertices Diagonal Triangle and
Orthogonal circumscribed DT-hyperbola
through vertices Diagonal Triangle and
QA-DT-Conic through:
Next table lists a number of self-involutary cubics.
QA-Cu1 till QA-Cu5:  QA-DT Cubics
self-involutary cubic
QA-Cu7: QA-Quasi Isogonal Cubic
self-involutary cubic
  • QA-Tf2(P)=DT-Tripole(QL-Tf2h(DT-Tripolar(P))), where QL-Tf2h=QL-Tf2 wrt the Quadrilateral formed by the QA-Tr1-trilinear polars of the QA-vertices (See QA-8 and Ref-34, QFG#1497,#1506 by Eckart Schmidt).
  • QA-Tf2(P) is the common point of the three versions of QL-Tf6(P). See Ref-34, Eckart Schmidt, QFG#2179.
  • QA-Tf2 is an isoconjugation of the diagonal triangle QA-Tr1 with fixed points in the vertices of the quadrangle. See Ref-34, Eckart Schmidt, QFG#2875.
  • QA-Tf2(X) is the common point for polars of X wrt circumconics of the quadrangle. See Ref-34, Eckart Schmidt, QFG#2855, #2856, #2875.
  • Let Si (i=1,2,3) be the vertices of the QA-Diagonal Triangle and Li be the corresponding sidelines of the QA-Diagonal Triangle. QA-Tf2(P) is the common intersection point of the 4th harmonic lines of P.Si wrt (Lj,Lk), where i,j,k are different indices from (1,2,3). See Ref-34, Bernard Keizer, QFG#2878.

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