QA-Cu1: QA-DT-P4 Cubic / Zirkularkurve

The QA-DT-P4 Cubic is described by Daniel Baumgartner und Roland Stärk (see Ref-16 page 6). It is more fully described in Ref-15, Eckart Schmidt: “Das Steiner-Dreieck von vier Punkten”. In both papers it is called the “Zirkularkurve” because it is a circular cubic meaning that the circular points at infinity lie on this cubic (see Ref-13, circular points at infinity).

In general:
QA-Cu1 is a pivotal isocubic (see Ref-13, Pivotal Isocubic) with given pivot P on QA-Cu1,
• where the Reference Triangle has vertices A,B,C, being 3 QL-Tf1-images of P (wrt the 3 Quadrilaterals coming from the 3 QA-Component Quadrigons QA-3QG1),
• isoconjugation (see Ref-13, Isoconjugation) swapping P and the isogonal center (QA-P4) of ABCP,
• with fixed points the points of tangency of P to QA-Cu1.
This is a consequence of Ref-15b Paragraph 5.
In particular:
• QA-Cu1 is a pivotal isocubic wrt the diagonal triangle, invariant under the isoconjugation with fixed points Pi and with pivot QA-P4.
• QA-Cu1 is a pivotal isocubic wrt the Miquel triangle, invariant under the isogonal conjugation with pivot the point at infinity of QA-L4.
• QA-Cu1 is a pivotal isocubic wrt reference triangle consisting of the 3rd intersections of QA-Cu1 and DT-sides, invariant to the isoconjugation with fixed point QA-P4 and QA-DT-vertices and with pivot QA-P41.
• QA-Cu1 is a pivotal isocubic wrt reference triangle consisting of the 3rd intersections of QA-Cu1 and the sides of the Miquel triangle, invariant under the isoconjugation with fixed points in the point at infinity of QA-L4 and the vertices of the Miquel triangle and with pivot the intersection point of QA-Cu1 and its asymptote.
See Ref-34, Eckart Schmidt, QFG messages #1657, #1658.
QA-Cu1 is a pK(QA-P16,QA-P4) cubic wrt the QA-Diagonal Triangle in the terminology of Bernard Gibert (see Ref-17b). (note Eckart Schmidt)

Equations:
Equation CT-notation:
a2 Ta y z (r y - q z) + b2 Tb x z (r x - p z) + c2 Tc x y (q x - p y) = 0
where:
Ta = a2 (p + q) (p + r) b2 p (p + q) c2 p (p + r)
Tb = b2 (q + r) (q + p) c2 q (q + r) a2 q (q + p)
Tc = c2  (r + p) (r + q) a2 r (r + p)  b2 r (r + q)

Equation DT-notation:
(c4 p2 q2+b4 p2 r2-a4 q2 r2-b2 c2 p2 (-p2+q2+r2)) (r2 y2-q2 z2) x
+(c4 p2 q2-b4 p2 r2+a4 q2 r2-a2 c2 q2 (p2-q2+r2)) (-r2 x2+p2 z2) y
+(-c4 p2 q2+b4 p2 r2+a4 q2 r2-a2 b2 r2 (p2+q2-r2)) (q2 x2-p2 y2) z = 0
1st coordinate Infinity Point CT-notation:
a2 - b2 p /(p + r) - c2 p /(p + q)
1st coordinate Infinity Point DT-notation:
p2 (-2 a2 q2 r2+b2 r2 (p2+q2-r2)+c2 q2 (p2-q2+r2))

Properties (most from Eckart Schmidt):
the vertices of the Reference Quadrangle,
the vertices of QA-Tr1 (QA-Diagonal Triangle),
the vertices of QA-Tr2 (QA-Miquel Triangle): the 3 QA-versions of QL-P1,
the incenter and 3 excenters of the Miquel Triangle,
the 3 QA-versions of QG-P16 (Schmidt Point),
QA-P3, the Gergonne-Steiner Point,
QA-P4, the Isogonal Center,
QA-P41, which is QA-Tf2(QA-P4),
QA-Tf1(QA-L9),
the circular points at infinity (so the cubic is called circular).
• The tangents at P1, P2, P3, P4 to QA-Cu1 are concurrent in the pivot QA-P4.
• The tangents at S1, S2, S3 and QA-P4 to QA-Cu1 are concurrent in QA-P41.
• The tangents at M1, M2, M3 to QA-Cu1 are concurrent on the cubic in the intersection with the asymptote.
• The asymptote of QA-Cu1 // QG-P1.QG-P16 // QA-P3.QA-P4 // QA-P1.QA-P6 = QA-L4.
• The vertices of the cevian triangle of QA-P4 wrt the QA-Diagonal Triangle lie on the cubic.
• Triangle M1.M2.M3 is perspective with the QA-Diagonal Triangle with Perspector QA-P3, which is the isogonal conjugate of QA-P4 wrt the Miquel Triangle.
• Every point on the cubic has its isogonal conjugate wrt the Miquel Triangle M1.M2.M3 also on the cubic. All connecting lines of points and its isogonal conjugates (wrt Miquel Triangle) concur in an infinity point Q1 of the asymptote. This makes the QA-DT-P4 Cubic also a Pivotal Isogonal Circular Cubic wrt the Miquel Triangle with pivot Q1. See Ref-14 and Ref-17b.
• Q2 is the intersection point of the cubic QA-Cu1 and its asymptote. It is the isogonal conjugate (wrt the Miquel Triangle) of the infinity point of the asymptote of the cubic. Q2 is also the 4th intersection point of the circumcircle of the Miquel Triangle and the cubic. Q2 is also the intersection point of the tangents at M1, M2, M3 to the cubic (see Ref-15c). Q2 is also the Reflection of QA-P9 in the circumcenter of the Miquel Triangle M1.M2.M3.
• QA-Cu1 is the locus of intersection points of QA-Tf2(L) and QL-Tf1(L), where L = variable line through QG-P1 (Ref-34, Eckart Schmidt ,QFG message #225).
• Let P12, P13, P14, P23, P24, P34 be the projections of P on P1P2, P1P3, P1P4, P2P3, P2P4, P3P4. QA-Cu1 is the locus such that lines P12P34, P13P24, P14P23 are concurrent. See [34], Seiichi Kirikami QFG # 1018.
• Let P12, P13, P14, P23, P24, P34 be the projections of P on P1P2, P1P3, P1P4, P2P3, P2P4, P3P4. The lines P-P23 and P-P14 are rotated about P of a certain angle t ≠ π/2. These intersect P2P3, P1P4 respectively at two points on a line, say L1. Define L2, L3 similarly. Then L1, L2, L3 concur iff P lies on the same cubic QA-Cu1 for any t (note Bernard Gibert, March 15, 2015).
• The pedal circles of some point P wrt BCD, CDA, DAB, ABC are concurrent at a point Q. The Simson lines of Q wrt the pedal triangles of P wrt BCD, CDA, DAB, ABC are concurrent if and only if P lies on QA-Cu1 (Ngo Quang Duong in Ref-34, QFG messages #1529,#1531,#1534).
• QA-Cu1 is an anallagmatic curve (a curve invariant under inversion). The circles of inversion are the circles with their center in the excenters of the Miquel Triangle QA-Tr2 intersecting orthogonal in pairs with radical center the incenter of QA-Tr2. See Ref-34, Eckart Schmidt, QFG message#1304.
• QA-P11.QA-P41 intersects QA-Co2 on QA-Cu1. See Ref-34, Eckart Schmidt, QFG-message #1666.
• The QA-Tf2-conic of the QA-Cu1-asymptote contacts QA-Cu1 in QA-P3. See Ref-34, Eckart Schmidt, QFG-message #1666.
• A QA-circumconic through QA-P4 contacts QA-Cu1 in this point. See Ref-34, Eckart Schmidt, QFG-message #1666.
• QA-circumconics intersect QA-Cu1 in two further points collinear with QA-P41. See Ref-34, Eckart Schmidt, QFG-message #1666.
• All pivotal isogonal isocubics wrt QA-Tr2 intersect QA-Cu1 in two QA-Tr2-isogonal conjugated points collinear with the pivot. See Ref-34, Eckart Schmidt, QFG-message #1666.
• QA-Cu1 is the locus of points P, whose QL-Tf1-images lie perspective with the Diagonal Triangle QA-Tr1 with perspector QA-Tf2(P). See [34], Eckart Schmidt, QFG-message #2337.
• QA-Cu1 is invariant under the QA-Tf16 transformation. See Ref-34, QFG#3277.

Vernieuwen