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This is a sequel to CL035 devoted to circular pKs with pole Ω and pivot P. When P is given, recall that Ω is the barycentric product P x igP and the orthic line (L) of the cubic is parallel to the line P, gcP. (L) passes through a given point Q if and only if P lies on a circular circumcubic K(Q) with singular focus F, passing through H and the infinite point of the line OQ. Denote by QF the mapping Q → F and by FQ its inverse. These are quadratic transformations given by : QF (x : y : z) = a^2 b^2 c^2 x^2+b^4 c^2 x ya^2 c^4 x y2 b^2 c^4 x y+c^6 x ya^2 b^4 x z+b^6 x z2 b^4 c^2 x z+b^2 c^4 x za^6 y z+a^4 b^2 y z+a^4 c^2 y z : : FQ (x : y : z) = a^2 (a^4 x^22 a^2 b^2 x^2+b^4 x^22 a^2 c^2 x^2+b^2 c^2 x^2+c^4 x^2+a^4 x y2 a^2 b^2 x y+b^4 x ya^2 c^2 x y+a^4 x za^2 b^2 x z2 a^2 c^2 x z+c^4 x z+a^4 y za^2 b^2 y za^2 c^2 y z) : : Properties • QF has three singular points which are O = X(3) and the imaginary foci M1, M2 of the MacBeath inconic whose real foci are O, H. These points M1, M2 are isogonal conjugates and lie on the perpendicular bisector of OH. They also lie on every cubic spK(M, X5) as in CL055 and, in particular, on the Napoleon cubic K005. • FQ has three singular points which are X(265) and the circular points at infinity. • The common fixed points are A, B, C and H. Construction of F Let (J) and (K) be the Jerabek and Kiepert hyperbolas. If Q ≠ O is a given point, let • Q1 be the second point of (J) on the line OQ. • Q2 be the second point of (K) on the line GQ. • Q3 be the intersection of the lines {Q1, X265} and {Q2, X94}. Q3 is in fact the barycentric product Q x X94. • A' be the second point of (O) on the line AQ3. Then F is the second point of the circle {A, A', X265} on the line {Q1, X265}. The construction of Q for a given finite point F is easily obtained by reversing these steps. Another description : reflect Q in the sidelines of ABC to obtain Qa, Qb, Qc respectively. The radical center of the circles with diameters AQa, BQb, CQc is F. Note that the circles concur at F if and only if Q lies on K1155 = pK(X6, X523) in which case F lies on Q155. A generalization is to be found below. *** QFimage of a curve Generally speaking, the QFimage of a curve (C) of degree n is a curve (C') of degree 2n passing through the singular points of FQ. If (C) contains m singular points of QF (counting multiplicity), the degree of (C') is reduced to (2n  m). Obviously, if (C) contains one (or several) fixed point(s), then (C') will contain the same point(s). QFimage of a line • the perpendicular bisector of OH (passing through M1, M2) is contracted into the point X(265). • a line passing through O is transformed into a line passing through X(265). For instance, the Brocard axis is transformed into the Fermat line. • a line not passing through O is transformed into a circle passing through X(265).For instance, the line at infinity is transformed into the Johnson circle C(H, R) which contains X(265). QFimage of a conic The QFimage of a conic is generally a bicircular quartic with little interest. We shall only mention a few special cases. • The isogonal transform of the perpendicular bisector of OH is the circumconic with perspector X(50), passing through X(54), X(110) and obviously M1, M2. Its QFimage must be the circumcircle (O) which contains X(1141) = QF(X54) and X(476) = QF(X110). • The QFimage of a rectangular circumhyperbola is generally a bicircular circumquartic passing through H and X(265) but when it is the Jerabek hyperbola, the quartic splits into the line at infinity and a circular nodal circumcubic with node X(265), passing through {X4, X115, X265, X1141, X13273, X14989, X17702}. QFimage of a circumcubic The QFimage of a circumcubic (K) passing through the three singular points O, M1, M2 of QF is a circular circumcubic (K') passing through X(265). Note that, when O is a node on (K), the cubic (K') must split into the line at infinity and a circumconic passing through X(265). For instance, K009 gives the Jerabek hyperbola. The most interesting case is obtained when (K) also passes through H which is the "last" focus of the MacBeath inconic and also the "last" fixed point of QF on (K). Indeed, (K) becomes a spK(P, X5) for some point P and (K') becomes a circular circumcubic passing through H and X(265). The following table shows a selection of remarkable cubics (K), (K') and P when (K) = spK(P, X5). 



Notes • When P lies on the line at infinity, (K) is an isogonal focal nK with root on the trilinear polar of X(264) and singular focus on (O). (K') is a focal cubic with singular focus H and both cubics share the same orthic line. See pink cells. The inverse of (K) in the circumcircle (O) is also a focal cubic with singular focus X(186). • (K') is also a focal cubic when P lies on a rectangular hyperbola with center X(5) passing through {3, 4, 195, 576, 1147, 2574, 2575, 2888, 2904}. The singular focus lies on K025. See orange cells. • spK(P, X5) is a nK if and only if P lies on the line at infinity as above or on the circumconic with center X(5), perspector X(216) passing through X(110), X(265), X(1625). In this latter case, spK(P, X5) is a nK with pole on cK(#X6, X4) and root on the circumconic with center X(216), perspector X(5). spK(X110, X5) and spK(X265, X5) are decomposed cubics and spK(X1625, X5) is nK(X879, X5 x X290, X2). • spK(P, X5) is a K0 if and only if P lies on the trilinear polar of X(925) and a psK if and only if P lies on K044. It follows that there are three spK(P, X5) which are pKs. See green cells where a red P denotes a pK. Any such pK or psK must have its (pseudo)pole on K176 and (pseudo)pivot on K674. The (pseudo)isopivot lies on pK(X54 x X184, X54), the isogonal transform of K674. • When P lies on the Euler line, spK(P, X5) is a member of a pencil of circumcubics passing through M1, M2 and X(3), X(4) both counted twice since the tangents (when defined) pass through X(74). See blue cells and also Table 54. This pencil is invariant under isogonal conjugation : the isogonal transform of spK(P, X5) is spK(P', X5) where P' is the reflection of P about X(5). The QFimage of this pencil is another pencil of circular circumcubics passing through X(4), X(30), X(265) with singular foci on the line X(4), X(94), X(143), X(146), etc.




QF generalized and related curves Let Q be a point with pedal triangle PaPbPc. Denote by QaQbQc the image of PaPbPc under the homothety h(Q, T), where T is a real number or infinity. Let F_{T} be the radical center of the circles with diameters AQa, BQb, CQc and then QF_{T} is the mapping Q → F_{T}. When T = 2 we find QF as above. QF_{T} has four fixed points namely A, B, C, H and three singular points Q1, Q2, Q3 on the cubic K028. If M_{T} is the image of X(74) under the homothety h(O, 1  2/T), these three points are the common points (apart M_{T}) of the circle (C_{T}) with center O passing through M_{T} and the rectangular hyperbola (J_{T}1) passing through O, H, M_{T}, X(2574), X(2575) which is homothetic to the Jerabek hyperbola. Its center Ω1 is the midpoint of HM_{T}. When T varies, the locus of F_{T} is a rectangular hyperbola H(Q) passing through H, Q, agQ. H(O) decomposes into the line at infinity and the Euler line. When Q ≠ O, H(Q) meets the line at infinity at the isogonal conjugates of the intersections of (O) and the line OQ. H(Q) also decomposes into two perpendicular lines when : • Q lies on (O) in which case H(Q) is the union of the Steiner line of Q and its perpendicular at Q. The center of H(Q) lies on a trifolium with node H. • Q lies on the line at infinity in which case H(Q) is the union of the line at infinity and a line passing through H. • Q lies on K028 in which case H(Q) is the union of two perpendicular lines secant on a circular unicursal circumsextic with singularity at H, passing through X(30), X(5176), X(39266), the infinite points of K003. H is a quintuple point with tangents parallel to the asymptotes of the Jerabek hyperbola and the Orthocubic K006. The asymptotes of this sextic are the line X(30)X(113), the parallels at X(3845) to the asymptotes of K003, two isotropic lines secant at X(10113). *** Let P be a fixed point. The locus of Q such that P, Q, F_{T} are collinear is a cubic K(P, T) = spK(P_{T}, Q_{T}) as in CL055. P_{T} is the image of the complement cP of P under the homothety h(O, T / (T  1)) and Q_{T} is the midpoint of P, P_{T}. In general, K(P, T) passes through nine points independent of T namely A, B, C, H, P, the three other points of pK(X6, P) on (O) and S on the line HP and on the line passing through O and the isogonal conjugate of the infinite point of the line HP. This defines a pencil of cubics for a given P when T varies. Special cases : • When T = 1 and P ≠ H, K(P, T) decomposes into (O) and the line HP. • When P = H, K(P, T) is K028 for any T ≠ 1. If T = 1, the locus of Q is the whole plane. In all other cases, K(P, T) also contains the following points that depend on T : • the infinite points of pK(X6, P_{T}). • the three singular points Q1, Q2, Q3 of QF_{T} on K028. • the isogonal conjugate gP_{T} of P_{T}. • the four foci of the inconic with center Q_{T}. Special cubics : • When T = 0 and P not on the line at infinity, K(P, T) is a circumstelloid with asymptotes parallel to those of K003. The radial center is X = h(X5, 1/3)(P). X lies on the cubic when P lies on a central stelloid passing through X(5), X(20), X(382) giving the central circumstelloids K026, K080, K525. This central stelloid is in fact the image of K026 under the homothety h(X5, 3). • When T = ∞, K(P, T) is psK(gcP, tcP, P) as in CL071. It becomes a pK when P lies on the line GK. • When P lies on the line at infinity, K(P, T) is a circular cubic passing through gP which decomposes for T = 0. For T = 2, it becomes a focal isogonal nK passing through O.


Additional remarks K(P, T) meets the circle (C_{T}) at Q1, Q2, Q3 already mentioned and three other points R1, R2, R3 on another rectangular hyperbola (J_{T}2) also homothetic to the Jerabek hyperbola. (J_{T}2) also passes through : • P, • the reflection N_{T} of M_{T} about O, • the intersection E_{T} of the line OX(74) and the parallel at P to the Euler line. Its center Ω2 is the midpoint of PN_{T}. 
