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X(1), X(2), X(3), X(20) excenters infinite points of K243 Q1, Q2, Q3 : vertices of the Thomson triangle T1, T2, T3, P1, P2, P3, S1, S2, S3, X described below 

Let pK(K, P) be an isogonal pivotal cubic with pivot P. There is one and only one point M such that the polar conic of M is a circle unless P is one of the three points T1, T2, T3 described below for which there is a line of points with circular polar conics. Let T be the transformation that maps P to M and let T' be the inverse transformation. See further details below. T and T' transform any line L passing through O into two rectangular hyperbolas H, H' having the same points at infinity and passing through X(2), X(20) respectively. H contains the singular points T1, T2, T3 of T which are the common points of C(O, 3R), K004, K077. H' contains the singular points Q1, Q2, Q3 of T' which are the vertices of the Thomson triangle i.e. the common points of C(O, R), K002, K078. When L rotates about O, • L and H meet at two points on K077, • L and H' meet at two points on K078, • H and H' meet at two finite points M1, M2 on K609 and the line M1M2 passes through a fixed point X. 

X = X(35237) is the intersection of the lines X(1)X(1406), X(3)X(1495), X(6)X(30) and lies on the Stammler hyperbola. It is the tangential of O in K609. K609 meets the Stammler hyperbola at the in/excenters, X(3), X and the Wallace hyperbola at the in/excenters, X(2), X(20). It follows that the coresidual of the in/excenters is O thus any diagonal rectangular hyperbola passing through the in/excenters meets K609 at two points collinear with O. See the construction below. K609 meets the circumcircle at the vertices Q1, Q2, Q3 of the Thomson triangle and three other points P1, P2, P3 lying on the rectangular hyperbola passing through X(3), X(20), X(74) and the infinite points of the Jerabek hyperbola. These points also lie on nK0(X6, R) with R = X(2)X(385) /\ X(4)X(32). K609 meets C(O, 3R) at T1, T2, T3 on K004, K077 and three other points S1, S2, S3 on the rectangular hyperbola passing through X(2), X(3), X(1351) and the infinite points of the Jerabek hyperbola. 

Notes :
*** K609 is a member of several pencils of cubics, for example those generated by • K243 and the union of the Stammler hyperbola and the line at infinity • K169 and the union of the Wallace hyperbola and the Lemoine axis. See a generalization below. 



The following construction of M = T(P) is given in the 1959 Mathesis paper "Cercles polaires dans les cubiques autoisogonales à pivot" by Deaux. Let A1B1C1 be the pedal triangle of P and let C1b, B1c be the reflections of C1, B1 about B, C respectively. The parallels through B, C to CC1b, BB1c meet AC, AB at Ba, Ca respectively. Let La be the line BaCa and let Lb, Lc the two other lines defined similarly. These three lines La, Lb, Lc concur at the requested point M. Remark : these lines are parallel when P lies on C(O, 3R) and they eventually coincide when P is one of the points T1, T2, T3. *** With P = x : y : z, the first coordinate of M is : a^2 (2 b^2 c^2 x (x + y + z)  b^2 c^2 x^2 + c^2 SC y^2 + b^2 SB z^2 + 4 ∆^2 y z), and conversely, with M = x : y : z, the first coordinate of P is : a^2 ( b^2 c^2 x^2 + c^2 SC y^2 + b^2 SB z^2  4 ∆^2 y z), where ∆ is the area of ABC. *** T fixes the in/excenters, swaps the circular points at infinity. It has three singular points namely T1, T2, T3. T maps • A, B, C to the midpoints of the altitudes of ABC, • the points at infinity of these altitudes to A, B, C, • the line at infinity to the circumcircle (O), • C(O, 3R) to the line at infinity, • the Darboux cubic K004 to the Thomson cubic K002, • the first Deaux cubic K077 to the second Deaux cubic K078, • any line not passing through T1, T2, T3 to a conic passing through Q1, Q2, Q3, • any line passing through O and not passing through T1, T2, T3 to a rectangular hyperbola passing through G, Q1, Q2, Q3, • the Euler line to the rectangular hyperbola passing through X(2), X(3), X(6), X(110), X(154), X(354), X(392), X(1201), X(2574), X(2575), X(3167), Q1, Q2, Q3. This is actually the Jerabek hyperbola of the Thomson triangle. 



Let S be a variable point on the circumcircle. The Steiner line (S) of S meets the line SX(376) at Q. The diagonal hyperbola (H) with center S through the in/excenters meets the line (L') through O and the isogonal conjugate Q* of Q at two points M1, M2 on K609. Note that the midpoint of M1M2 lies on the line SQ. K609 meets pK(X6, Q) at the four in/excenters, M1, M2 and three other collinear points on the line (L) which is the trilinear polar of X(110)÷S (barycentric quotient). (L) is tangent to the Kiepert parabola. These three points are not always real and are not visible in the opposite figure. It follows that K609 belongs to each pencil of cubics generated by one pK(X6, Q) with Q on K616 and the decomposed cubic which is the union of (L) and (H). With Q = X(69), X(376) we obtain the two special cases mentioned above namely K169, K243. When S traverses the circumcircle, the locus of Q is the nodal cubic K616. 
