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X(2), X(3), X(15), X(16), X(30), X(110), X(5463), X(5464)

foci of the Steiner in-ellipse

Q1, Q2, Q3 vertices of the Thomson triangle

R1, R2, R3 traces of the perpendicular bisector of X(2)X(110) on its sidelines

more points and larger figure below

K463 is the locus of contacts of tangents drawn through X(110) – the focus of the Kiepert parabola – to the circles passing through G and O.

Thus, K463 is a focal cubic with singular focus X(110). Its axis (or orthic line) is the Euler line hence its real asymptote is its parallel at X(74). This is also the real asymptote of the Neuberg cubic K001.

The polar conic of X(110) is the circle through G, O, X(110), X(842), tangent at O to the Brocard axis.

K463 is the Laplacian of Q037. This is the locus of M whose polar conic in Q037 is a rectangular hyperbola.

The inverse in the circumcircle of K463 is K912.

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K463 is a member of the pencils generated by :

• two decomposed cubics : one is the union of the Euler line and the circumcircle, the other is the union of the line at infinity and the Thomson-Jerabek hyperbola. This pencil also contains K834, K913 and K1095.

• the Neuberg cubic and the decomposed cubic which is the union of the Brocard axis and the line at infinity counted twice. This pencil also contains K894.

• the Neuberg strophoid K725 and the decomposed cubic which is the union of the line X(3)X(110) and the line at infinity counted twice.

See also K1286 and Table 81 for other general properties.

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K463 is invariant under

• the isogonal conjugation with respect to the Thomson triangle. It is the focal cubic K187 for this latter triangle i.e. the locus of foci of inscribed conics with center on the common Euler line of ABC and Q1Q2Q3.

• the isogonal conjugation with respect to the triangle T with vertices X(2), X(15), X(16). More precisely, K463 is a nK in T and meets the sidelines of T on the line X(3), X(67), X(542), etc.

• (more generally) the isogonal conjugation with respect to any triangle T with vertices X(2) [resp. X(3)] and the two remaining points of K463 which lie on a line passing through X(3) [resp. X(2)]. For any point M on the cubic, the isogonal conjugate M* of M with respect to ANY triangle above (including the Thomson triangle) does not depend of the choice of the triangle.

• the Psi transformation which is the product of the reflection about one axis of the Steiner inellipse and the inversion with circle that of diameter F1F2, the foci of the ellipse. See "Orthocorrespondence and Orthopivotal Cubics", §5 and K018, K022, also K508, K816. K463 is a Psi-cubic as in Table 60. Note that, for M on K463, Psi(M) coincide with M* as above.

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Other points on K463

• X, the tangential of X(110), on the line X(30)X(74) and on the tangent at X(110) to the circle X(2)X(3)X(110),

• Y on the line X(30)X(110) and on the perpendicular bisector of X(2)X(3),

• X' = X(2)X /\ X(3)Y and Y' = X(2)Y /\ X(3)X.

It follows that K463 is an isogonal nK with respect to each triangle OXX', OYY', GXY', GX'Y.

Common points with K001:

K463 meets the Neuberg cubic at X(30), circular points at infinity J1, J2 (these are three double points), O, X(15), X(16).

Common points with K002 :

K463 meets the Thomson cubic at O, G, Q1, Q2, Q3 (on the circumcircle) and the four foci of the Steiner inellipse. The real foci are F1 and F2 on the figure. See also K727, K758, K833.

Common points with K005 :

K463 meets the Napoleon cubic at O (triple point) and the centers Ae, Be, Ce, Ai, Bi, Ci of the equilateral triangles erected on the sides of ABC, either externally or internally.

Note that K005 and K463 have the same tangent – the line O, X(74), X(110) – and the same polar conic at O. This latter conic is the rectangular hyperbola with center G, with asymptotes parallel to those of the Jerabek hyperbola, passing through O, K, X(381), X(599) and the four foci of the Steiner inellipse. It follows that the tangents at these foci pass through O.

Common points with K018 :

K463 meets the second Brocard cubic at G, X(15), X(16), J1, J2 and the four foci of the Steiner inellipse.

Common points with K078 :

K463 meets the stelloid K078 at O, G, Q1, Q2, Q3 and the four foci of the conic with center O inscribed in the Thomson triangle. See K804 for further details.

Common points with K725 :

K463 meets the Neuberg strophoid K725 at X(110) and four double points namely X(3), X(30), J1, J2.