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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 circum-cubic 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 y-a^2 c^4 x y-2 b^2 c^4 x y+c^6 x y-a^2 b^4 x z+b^6 x z-2 b^4 c^2 x z+b^2 c^4 x z-a^6 y z+a^4 b^2 y z+a^4 c^2 y z : :

FQ (x : y : z) = a^2 (a^4 x^2-2 a^2 b^2 x^2+b^4 x^2-2 a^2 c^2 x^2+b^2 c^2 x^2+c^4 x^2+a^4 x y-2 a^2 b^2 x y+b^4 x y-a^2 c^2 x y+a^4 x z-a^2 b^2 x z-2 a^2 c^2 x z+c^4 x z+a^4 y z-a^2 b^2 y z-a^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.

***

QF-image of a curve

Generally speaking, the QF-image 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).

QF-image 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).

QF-image of a conic

The QF-image 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 circum-conic with perspector X(50), passing through X(54), X(110) and obviously M1, M2. Its QF-image must be the circum-circle (O) which contains X(1141) = QF(X54) and X(476) = QF(X110).

• The QF-image of a rectangular circum-hyperbola is generally a bicircular circum-quartic 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 circum-cubic with node X(265), passing through {X4, X115, X265, X1141, X13273, X14989, X17702}.

QF-image of a circum-cubic

The QF-image of a circum-cubic (K) passing through the three singular points O, M1, M2 of QF is a circular circum-cubic (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 circum-conic 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 circum-cubic 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 circum-conic 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 circum-conic 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 circum-cubics 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 QF-image of this pencil is another pencil of circular circum-cubics 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_T1) 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 circum-sextic 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 circum-stelloid 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 circum-stelloids 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_T2) also homothetic to the Jerabek hyperbola.

(J_T2) 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.