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K1093

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X(2), X(6), X(9741), X(21448)

vertices of the Thomson triangle

foci of the orthic inconic (K-ellipse when ABC is acute)

other points below

Geometric properties :

K1093 = spK(X1992, X6) is the isogonal transform of K314 = spK(X2, X6) as in CL055.

It is a member of the pencil generated by K018 and K102. See Table 52.

Any nK0(X6, P) with P on the orthic axis is a focal cubic with singular focus on the circumcircle. The pencil generated by this nK0 and its Hessian cubic always contains a stelloid whose Hessian cubic is the nK0 itself. See CL025.

When P is not on the orthic axis, the pencil generated by the nK0 and its Hessian cubic does not generally contain a circular cubic nor an equilateral cubic unless P lies on K1093 in which case it contains both of them.

Special case : K024 = nK0(X6, X6) is a stelloid whose Hessian cubic is the focal cubic K193.

Other properties of K1093

• K1093 meets the line at infinity at three points lying on pK(X2, X11185), pK(X3, X7493), pK(X6, X1992), K294 = pK(X574, X2), pK(X599, X69), pK(X5094, X4). Note that, for any pole Ω on K294, there is a pivot P on pK(X2, X11185) such that pK(Ω, P) shares the same points at infinity as K1093.

• K1093 meets the circumcircle (O) again at the vertices Q1, Q2, Q3 of the Thomson triangle and the tangents at these points concur at X(6). See K1094 for a generalization. The polar conic of X(6) is the Thomson-Jerabek hyperbola. Note that this hyperbola is the polar conic of X(376) in K003, also the polar conic of X(3) in K832.

• K1093 passes through R1, R2, R3 which are the feet of the altitudes of Q1Q2Q3 whose orthocenter is X(2). It follows that K1093 is a pK with pivot X(2), isopivot X(6) in the Thomson triangle.

• K1093 meets the Steiner ellipse (S) again at S1, S2, S3 which also lie on pK(X2, S) where S is the point with barycentics a^4-3 a^2 b^2-3 a^2 c^2+2 b^2 c^2 : : , SEARCH = -0.243010233646405, on the lines {2,2418}, {3,194}, {6,99}, {193,376}, {148,381}, etc. S = X(31859) in ETC.

• K1093 meets the sidelines of ABC again at A1, B1, C1. These points are on the parallels at X(2) to the cevian lines of X(1992).

• K1093 meets the orthic axis of ABC again at A2, B2, C2. These points are on the medians of ABC. The tangents to K1093 at these points concur at X(11284) on the Euler line.

• K1093 meets the symmedians of ABC again at A3 = SA : 2b^2 : 2c^2 and B3, C3 likewise. These points are the isogonal conjugates of the reflections in X(6) of the vertices of the pedal triangle of X(6).

• K1093 meets K314 at A, B, C, X(2), X(6) and the foci of the orthic inconic, the inconic with center X(6).

K1093a