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X(2), X(4), X(5), X(52), X(193), X(343), X(5392), X(27364)

vertices of the orthic triangle

other points below

note : X(5392), X(27364) are the sixth common points of K1087 and the Kiepert hyperbola, the rectangular circum-hyperbola through X(5) respectively

Geometric properties :

K1087 is the isogonal pK with pivot X(343) with respect to the orthic triangle. It is the locus of M such that X(343), M and the isogonal conjugate M* of M with respect to the orthic triangle are collinear. K1087 is then a member of the Euler orthic pencil. Recall that M* is also the X(4)-Ceva conjugate of M.

The tangential Q of X(343) is the isopivot (with respect to the orthic triangle). It is the X(4)-Ceva conjugate of X(343) in ABC with SEARCH = 3.29121185555526.

(S) is the circum-conic with perspector X(343). It meets K1087 at S1, S2, S3 such that the orthic inconic (C) is inscribed in the triangle S1S32S3.

The contacts R1, R2, R3 of (C) with the triangle S1S32S3 also lie on K1087.

K1087 meets the line at infinity at the same points as pK(Ω, P) where Ω lies on K612 = pK(X216, X2) and P lies on K045 = pK(X2, X264). This is the case of K045 and K612 themselves and also pK(X3, X3), pK(X6, X1993), pK(X343, X69), pK(X2165, X5392), etc.

K1087 meets the circumcircle at the same points Q1, Q2, Q3 as pK(X6, Po) where Po is a point on the lines {22, 2393}, {51, 110}, {52, 539}, etc. These two cubics meet again on the line X(6), X(1147). Po = X(27365) in ETC now (2018-11-07).

K1087 meets the Steiner ellipse at the same points as pK(X2, Ps) where Ps is a point on the lines {4, 193}, {6, 297}, {30, 3164}, {53,648}, {69, 458}, etc. These two cubics meet again at X(2), X(193), X(343}. Ps = X(27377) in ETC now (2018-11-07).

The isotomic transform of K1087 is pK(X95, X275).