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X(2), X(376), X(1992), X(2094), X(3241), X(3543), X(6172), X(39158), X(39159), X(39160), X(39161), X(60874), X(60875), X(60876)

infinite points of K002 and K007

A', B', C' : reflections of A, B, C in G

A", B", C" : reflections of G in A, B, C

Geometric properties :

See CL045 for related definitions and examples.

Let Q = u : v : w be point distinct of G. The locus of M such that Q, M and the Tripolar Centroidal Conjugate TCC(M) of M are collinear is a nodal circum-cubic K(Q) with node G.

K(Q) meets the sidelines of ABC at U, V, W respectively. U = 0 : 2u + v : 2u + w and V, W are defined cyclically.

If Q' is the reflection of Q in G and if S is its isotomic conjugate, then U is the point on BC that lies on the parallel at G to AS and on the parallel at Q to AQ'.

UVW is the pedal triangle of some point T if and only if Q lies on K1360. In this case, T lies on K758, the Thomson isogonal transform of K002. Recall that K758 contains the foci X(39162), X(39163), X(39164), X(39165) of the Steiner in-ellipse.

K295 is one of these cubics obtained when Q = X(1992), then T = G.

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K1360 is the reflection in G of the Lucas cubic K007, also the transform of K002 under the homothety h(G, 2). Note that the three cubics have the same tangent at G which passes through K.

K1360 meets K007 at three infinite points, G (twice) and the four foci X(39158), X(39159), X(39160), X(39161) of the Steiner ellipse.

K1360 meets K002 at three infinite points, G (twice) and four points that lie on the transform (H) of the Kiepert hyperbola under the homothety h(G, -3).