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X(4), X(30), X(143), X(1147), X(1992), X(2574), X(2575), X(15471) isogonal conjugates X(15460) = X(1312)*, X(15461) = X(1313)* traces of the line (L) through X(107), X(110) traces of the circle (C) through X(107), X(110), X(125) 

Geometric properties : 

We meet K934 in Hyacinthos #26858 (A. P. Hatzipolakis) and more specifically here (Angel Montesdeoca, in Spanish). A generalization is given below. K934 is the locus of the barycentric product N = M x M^{⊥} when M traverses the Euler line (in which case the orthocorrespondent M^{⊥} of M traverses the line GK). Note that N is also the midpoint of M and H/M, the HCeva conjugate of M. See a list of pairs {M, N} at the bottom of this page. See also Table 64, Lemoine cubics. K934 is a cuspidal cubic with cusp H and cuspidal tangent passing through X(107) and the center X(125) of the Jerabek hyperbola (J). K934 has three real asymptotes : one is the parallel to the Euler line at the center X(5972) of (C), the two others are the parallels at X(11746) to those of (J). K934 has one real point of inflection F = X(15471) on the line HK, the barycentric product of X(468) and X(1992). Note that X(468)^{⊥} = X(1992). 


Additional properties 

Let M, M' be two orthoassociate points on the Euler line (i.e. inverse in the polar circle). Let N, N' be the corresponding barycentric products as defined above. When M traverses the Euler line, • the line NN' envelopes the conic (E) inscribed in ABC with center X(5972). Note that the circle (C) is the orthoptic circle of the conic (E). • the lines MN and M'N' are parallel and tangent to (E), • the lines MN' and M'N meet on the hyperbola (H). *** (E) is tangent at H to the Euler line, at X(69) to the line GK. (E) also contains X(1092), X(1974), X(3043) and X(1312)*, X(1313)* with tangents passing through X(110) and X(1114), X(1113) respectively. It is then clear that (E) and (J) meet at four known points namely X(4), X(69), X(1312)*, X(1313)*. The triangles X(110)X(1113)X(1114) and X(4)X(1312)*X(1313)* are perspective at a point Z = X(15472) in ETC (20171206). Z lies on the lines {3,1112}, {4,110}, {6,74}, {25,1511}, {265,427}, {578,2777}, etc. *** (H) passes through X(4) with tangent the Euler line, X(182) with tangent passing through X(378), X(1312)*, X(1313)*, X(1593)*. Note that the (not represented) tangents to (H) at X(1312)*, X(1313)* meet at Z as above and pass through X(1114), X(1113) respectively. The center L of (H) is now X(15473) in ETC (20171206). L lies on the lines {4, 74}, {25,113}, etc. 



Generalization 

Let (L) be a line passing through the orthocenter H of ABC, with trilinear pole P, meeting the sidelines of ABC at U, V, W respectively. For any point M on (L), let N = f(M) = M x M^{⊥} = midpoint of M and H/M as defined above. When M traverses (L), the locus of N is a cuspidal cubic (K) with cusp H. (K) meets the sidelines of ABC at • three collinear points A', B', C' on (L') which are the fimages of U, V, W respectively. A' is the midpoint of U and AP /\ BC. • six concyclic points A1 and A2 on BC, B1 and B2 on CA, C1 and C2 on AB, defined as follows. (L) meets the circle with diameter BC at two points which are orthogonally projected on BC at A1 and A2. Let (C) be the circle through these six points whose center is denoted Ω. For any line (L), the line (L') is tangent (at S) to the circumconic with perspector X(393), the barycentric square of H. The line HS is the cuspidal tangent at H. Now, if M and M' are two orthoassociate points on (L) and if N, N' are their fimages then the line NN' envelopes the inscribed conic (E) with center Ω. This conic passes through H with tangent (L). Here again, the lines MN and M'N' are parallel and tangent to (E). Note that (K) and (E) meet at H and are bitangent at two other points T1, T2. The common tangents at these points meet at T on the circumcircle (O). T is the antipode of the isogonal conjugate of the infinite point of (L). Remark : when (L) rotates around H, the intersection of (L) and (L') lies on K406 (Angel Montesdeoca).




A list of pairs {M, N = f(M) = M x M^{⊥}} borrowed from Angel Montesdeoca's web site 

{1, 56}, {2, 1992}, {3, 1147}, {5, 143}, {11, 513}, {13, 11080}, {14, 11085}, {15, 11136}, {16, 11135}, {46, 56}, {52, 143}, {113, 30}, {114, 511}, {115, 512}, {116, 514}, {117, 515}, {118, 516}, {119, 517}, {120, 518}, {121, 519}, {122, 520}, {123, 521}, {124, 522}, {125, 523}, {126, 524}, {127, 525}, {128, 1154}, {131, 13754}, {132, 1503}, {133, 6000}, {136, 924}, {137, 1510}, {155, 1147}, {193, 1992}, {371, 32}, {372, 32}, {487, 6337}, {488, 6337}, {1312, 2575}, {1313, 2574}, {1560, 2393}, {1566, 926}, {1785, 1875}, {1845, 1875}, {2039, 3413}, {2040, 3414}, {2679, 804}, {2902, 11136}, {2903, 11135}, {3258, 526}, {3259, 900}, {5099, 690}, {5139, 3566}, {5190, 8676}, {5509, 814}, {5510, 3667}, {5511, 3309}, {5512, 1499}, {5513, 674}, {5514, 3900}, {5515, 834}, {5516, 6085}, {5517, 8678}, {5518, 4083}, {5519, 6084}, {5520, 8674}, {5952, 8702}, {6110, 1990}, {6111, 1990}, {9151, 888}, {9152, 5969}, {9193, 9023}, {10017, 8677}, {12494, 8704}, {13234, 3849}. Note that f maps the nine point circle on to the line at infinity. 
