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X(2), X(3), X(5373), X(11472), X(39162), X(39163), X(39164), X(39165)

vertices of the Thomson triangle, excentral of Thomson, orthic of Thomson

infinite points of K243

points on (O) and nK0(X6, X7736) other than A, B, C

further details below

Geometric properties :

K1259 is the Orthocubic K006 of the Thomson triangle T. See Table 81 for general properties.

It is the locus of M such that X(2), M and the isogonal conjugate of M with respect to T are collinear. Recall that X(2) is the orthocenter of T.

Points on K1259 :

• obviously the centers of K006 evaluated with respect to T, in particular X(2), X(3), X(5373), X(11472) which are respectively X(4), X(3), X(1), X(155) in T.

• the foci X(39162), X(39163), X(39164), X(39165) of the Steiner inellipse since these points are two by two collinear with X(2) and isogonal conjugates with respect to both triangles ABC and T.

• vertices Q1,Q2,Q3 of the Thomson triangle T, with tangents concurring at O. Note that the polar conic of O is the Jerabek-Thomson hyperbola passing through these three points and X(2), X(3), X(6), X(110), X(154), X(354), X(392), X(1201), X(2574), X(2575), X(3167), etc.

• vertices I1, I2, I3 of the excentral triangle of T with tangents concurring at G. Note that the polar conic of G is a rectangular hyperbola passing through X(2), X(3413), X(3414), X(5373), I1, I2, I3, with center X(98).

• vertices H1, H2, H3 of the orthic triangle of T with tangents concurring at X(11284), a point on the Euler line.

• S1, S2, S3 other points on (O) and on nK0(X6, X7736). These points are the antipodes on (O) of the common points T1, T2, T3 (apart A, B, C) of (O) and K243 = pK(X6, X376).

• M1, M2, M3 midpoints of triangle S1S2S3.

• infinite points of K243. Note that X1259 and K243 share the same asymptotes. Their remaining common points are O and two imaginary points on the Brocard axis and on the circum-hyperbola passing through X(2), X(74), X(378) with perspectorX(8675). This is the isogonal transform of the line X(6), X(30).