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∑ SA x (b^2 c^2 x^2  a^4 y z) = 0 

X(6), X(523), X(1316), X(3569), X(5000), X(5001) circular points at infinity isogonal of Steiner infinite points isogonal of the MacBeath circumconic infinite points other points and more detailed figures below 

Geometric properties : 

K1091 is a focal cubic with focus X(6) passing through the Walsmith point X(5000). The polar conic of X(6) is the circle with diameter X(1344)X(1345). The real asymptote is the perpendicular at X(3569) to the Euler line. X(3569) is the tangential of X(6) and the circle with center X(2492) on the orthic axis, diameter X(6)X(3569), passes trhough X(5000), X(5001). K1091 is the Hessian cubic of a very simple stelloid with radial center X(6) and asymptotes parallel to those of K024. This stelloid has equation ∑ b^2 c^2 x (SA x^2 + a^2 y z) = 0, analogous to that of K1091, showing that these two cubics share 9 points of inflexion on the sidelines of ABC. If Ao, Bo, Co are the traces of the trilinear polar of X(25) on the sidelines of ABC then these points of inflexion are the cube roots of Ao, Bo, Co. Obviously only three of them A', B', C' are real, each lying on one sideline of ABC. K1091 meets the circumcircle (O) of ABC at three pairs of (not always real) points lying on the trilinear polars of X(2), X(4), X(6) i.e. on the line at infinity, the orthic axis, the Lemoine axis. These are the points mentioned above and more precisely : • O1, O2 are the circular points at infinity, collinear with X(523), • O3, O4, on the Lemoine axis, are the isogonal conjugates of the Steiner infinite points, collinear with X(3569), • O5, O6, on the orthic axis which is the orthic line of K1091, collinear with X(523), are the common points of (O), the nine points circle, etc. O5, O6 are real when ABC is obtusangle in which case X(5000), X(5001) are complex conjugates and K1091 is unipartite. Otherwise X(5000), X(5001) are real and K1091 is bipartite. K1091 meets the (imaginary) asymptotes of the Steiner ellipse at three pairs {Z1,Z2}, {Z3,Z4}, {Z5,Z6} of (not always real) points lying on the trilinear polars of X(13), X(14), X(523) respectively. More precisely : • Z1, Z2 are collinear with X(523). They also lie on K018 and the circumconic with perspector X(15), • Z3, Z4 are collinear with X(523). They also lie on K018 and the circumconic with perspector X(16), • Z5, Z6 are collinear with X(3569). They also lie on the Kiepert hyperbola and on any pK(X115, P) where P is a point on the trilinear polar of X(523). Indeed, their barycentric product (and also their midpoint) is X(115). They are in fact the barycentric products by X(523) of the Steiner infinite points. *** The polar conic of X(523) is the rectangular hyperbola passing through X(30), X(395), X(396), X(468), X(523), etc. It meets K1091 at four finite points which are the centers of anallagmaty of the cubic. These points lie on the parallels at X(6) to the asymptotes of the Jerabek hyperbola which pass through X(1344), X(1345) mentioned above. These lines are the bisectors at X(6) in triangle (T) = X(6)X(5000)X(5001) hence the four points are its in/excenters. K1091 is the isogonal pK with pivot X(523) with respect to (T) whose circumcircle contains X(3569), the tangential of X(6), and also X(111), X(112), X(115), X(187), X(1560), X(2079), etc. This circle is the antiorthocorrespendent of the line X(6), X(110), X(111), etc. The isogonal conjugates in (T) of X(5001), X(5000) are P7, P8 on K1091 and then P7 = X(6)X(5000) /\ X(523)X(5001) and P8 = X(6)X(5001) /\ X(523)X(5000). The line P7P8 contains X(98), X(648), X(2452), X(2967), etc. The midpoint of P7, P8 lies on the orthic axis. Note that K1091 is also a pK with pivot X(3569) in triangle with vertices X(1316), P7, P8. *** Locus properties 1. A circle passing through X(5000), X(5001) with center Ω1 on the orthic axis meets the line passing through X(6) and Ω1 at two points of K1091. Note that this circle is orthogonal to any circle of the pencil containing (O) and the nine points circle. See a generalization below. 2. The contacts of tangents drawn through X(6) to a circle (with center Ω2 on the Euler line) of the pencil above are two points of K1091. The line through X(6) and Ω2 meets the circle at two points on K1092, another focal cubic with focus X(6) passing through X(6), X(30), X(1379), X(1380), X(5000), X(5001), X(11472), X(16303), O5, O6. This is the isogonal pK with pivot X(30) in triangle with vertices X(6), O5, O6. *** Bipartite and unipartite K1091

ABC is acutangle and K1091 is bipartite. It is an isogonal pK in the (yellow) real triangle (T) = X(6)X(5000)X(5001) and the four centers of anallagmaty are also real. They lie on the parallels at X(6) to the asymptotes of the Jerabek hyperbola and also on the (green) rectangular hyperbola which is the polar conic of X(523). This hyperbola passes through X(30), X(395), X(396), X(468), X(523). Its center lies on the real asymptote of K1091 and it is the intersection of the lines {X6,X30}, {X50,X112}, {X53,X403}, etc, SEARCH = 0.0591384993287504. 

ABC is obtusangle and K1091 is unipartite. Two centers of anallagmaty are real and two are complex conjugates. 



Generalization Let P be a finite point not lying on the orthic axis (L), M a variable point on (L), (C) the circle with center M orthogonal to (O) hence passing through X(5000), X(5001). The line PM meets (C) at two points lying on the focal cubic K(P) with singular focus P, passing through X(523), X(5000), X(5001), the common points O5, O6 of (O) and (L), the common points P1, P2 of (O) and the polar line L(P) of P in (O). L(P) is also the trilinear polar of the cevapoint of X(6) and P.


Note that the third (always real) point T of K(P) on L(P) lies on the real asymptote of K(P). T is the tangential of X(523) in K(P). It is the antipode of P on the circle passing through P, X(5000), X(5001) hence orthogonal to (O). K(P) meets the Euler line at X(5000), X(5001) and P0, the orthogonal projection of P on the Euler line. The polar conic of P is the circle passing through P, O5, O6 hence with center on the Euler line. The polar conic of X(523) is the rectangular hyperbola (H) with center Ω on the parallel at P to the Euler line and on the real asymptote, the perpendicular at T to the Euler line.
Special case : when P lies on the Euler line, K(P) is an axial cubic symmetric in the Euler line. 

K(P) is the Hessian cubic of a stelloid S(P) with radial center P and asymptotes parallel to those of K024 when P lies on the rectangular hyperbola (H1) passing through X(6), X(74), X(110), X(113), X(125), X(1495), X(2574), X(2575), X(2931), X(3569), X(3580), X(5000), X(5001), X(7699), X(7703), X(10117), X(11472), X(15904), etc, hence homothetic to the Jerabek hyperbola (J). On the other hand, these asymptotes are parallel to those of K003 when P lies on another rectangular hyperbola (H2) meeting the line at infinity at the same points as the rectangular circumhyperbola passing through X(110). (H1) and (H2) have the same center X(468) and pass through X(5000), X(5001), O5, O6 where they are orthogonal. The asymptotes of each one are the axes of the other.


Reasonably simple centers on (H1) Z1 on the lines {X3,X6}, {X5,X1531}, {X23,X74}, {X26,X1204}, {X30,X125}, {X110,X186}, {X113,X468}, etc Z2 on the lines {X4,X6}, {X23,X146}, {X30,X110}, {X74,X468}, {X113,X858}, {X125,X403}, etc Z3 on the lines {X6,X647}, {X74,X512}, {X110,X250}, {X113,X525}, {X125,X523}, etc Z4 on the lines {X6,X468}, {X23,X69}, {X30,X599}, {X67,X74}, {X110,X524}, {X113,X511}, {X125,X2393}, etc Z5 on the lines {X6,X373}, {X74,X511}, etc Z6 on the lines {X69,X74}, {X110,X193}, {X113,X1596}, {X125,X126}, etc
(H1) is now called Walsmith rectangular hyperbola and some centers (including those above) were added to ETC as X321104 and X3211927 (20190420). 



Walsmith rectangular hyperbola and Walsmith triangle The Walsmith triangle WT has Avertex the intersection Wa of the Asymmedian of ABC and the perpendicular at X(125) to BC. It is inscribed in the Walsmith rectangular hyperbola (W) = (H1) above. The circumcircle (Cw) of WT passes through X(6), X(23), X(1316) and its circumcenter is Ow = X(32217) = a^2 (2 a^62 a^2 b^4+4 a^2 b^2 c^2b^4 c^22 a^2 c^4b^2 c^4) : : , SEARCH = 1.03032901140499, on the lines {X6,X23}, {X30,X182}, {X110,X524}, etc. This circle has diameter X(6)X(23) and Ow is the midpoint of X(6)X(23). The reflection X(32224) of X(1316) in Ow lies on (Cw) and on the lines {X6,X30}, {X23,X385}, etc, SEARCH = 3.73617798557606. Obviously, the orthocenter of WT lies on (W). This is the point Z4 above. The reflection of WT in S is the pedal triangle A'B'C' of X(23) where S = X(32223) is the point with coordinates 2 a^63 a^2 b^4+b^6+4 a^2 b^2 c^2b^4 c^23 a^2 c^4b^2 c^4+c^6 : : , SEARCH = 0.218352591903293, on the lines {X2,X3098}, {X23,X125}, {X69,X1974}, {X74,X1533}, etc. 

WT is orthologic (at X6 and X15303) and parallelogic (at X23 and X125) to the pedal triangle of X(2) hence these two triangles are inversely similar. More generally, WT is orthologic to the pedal triangle pedP of any P on the Euler line. The orthologic center O1 of WT to pedP lies on (W) and the orthologic center O2 of pedP to WT lies on the line passing through X(468), X(524). The line PO2 envelopes the parabola with focus X(112), directrix the line passing through X(110), X(525), X(935). This parabola is tangent at X(10295) to the Euler line and its point at infinity is X(1503). On the other hand, the line PO1 envelopes a very complicated quartic passing through X(1531), X(10295). 

WT is orthologic to the cevian (resp. anticevian) triangle of any P on K008 (resp. K043) and in both cases one of the orthologic centers is a point on K042. These are the Droussent, Droussent medial, Droussent central cubics. The other orthologic center lies on a complicated cubic passing through X(110), X(125). WT is orthologic to the antipedal triangle apdP of any P on the Jerabek hyperbola (J). The orthologic center O1 of WT to apdP lies on (W) and the orthologic center O2 of apdP to WT lies on the reflection (J') of (J) about X(6593), the midpoint of X(6), X(110).


In particular, WT is orthologic to the (green) antipedal triangle A'B'C' of X(67) and these two triangles are actually homothetic at the intersection Q = X(32227) of the lines {X2,X265}, {X25,X113}, {X110,X468}, etc, SEARCH = 1.36518782551494. The orthologic centers are the orthocenters of WT and apd(X67). These points are O1 = Z4 as above and O2, the reflection of X(5486) in X(6593). Note that the homothety that sends A'B'C' on to WT also transforms X(110) into X(468) and O2 into O1. 

Additional properties • WT is directly similar at X(1316) to the 2nd Brocard triangle and also to the 2nd orthosymmedial triangle. • WT is orthologic at O1 and X(6) to the 4th Brocard triangle where O1 = X(32225) is the barycentric product X(381) x X(524). • WT is orthologic at X(32226) and X(16176) to the reflection triangle. • WT is parallelogic at P1 and X(74) to the circummedial triangle where P1 is the reflection X(32224) of X(1316) in Ow mentioned above. • WT is orthologic to the circumcevian triangle ccvP of any P on the Walsmith circular quartic Q152.

