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X(7), X(9), X(144)

vertices A', B', C' of the cevian triangle of X(8)

their reflections A", B", C" in X(9), on the cevian lines of X(7)

reflections A9, B9, C9 of A, B, C in X(9)

the tangentials Z1 and Z2 of X(144) and X(7)

M1 and M2 on the focal axis of the Mandart ellipse

see below for further details

Geometric properties :

K1433 is a central cubic with center X(9) and inflexional tangent passing through X(55).

It is the transform of K443 = psK(X3, X69, X3) under the symbolic substitution SS{a → √a}.

Its pseudo-isopivot is X(1) hence the tangents at A, B, C are the internal bisectors.

K1433 meets the line {9, 2590, 3307), focal axis of the Mandart ellipse, at M1 and M2 which lie on K977 and on the circum-hyperbola (H) passing through X(7) and the isogonal conjugate of X(2590). The center of (H) lies on the lines {2, 7}, {11, 2447}. (H) is the isogonal transform of the line {55, 2590}.

The tangential Z1 of X(144) is the baryentric quotient X(9) ÷ X(31527), a point on the line {144, 200}

The tangential Z2 of X(7) is the X(8)-Ceva conjugate of X(19605), a point on the line {7, 1699}.

The pairs of points {M1, M2} and {Z1, Z2} are symmetric about X(9).

K1433 meets the line at infinity at three points which lie on a group of pKs with pole on K259, pivot on the anticomplement of K220, isopivot on K220.

Examples : K1082 = pK(X1, X7), pK(X3, X347), pK(8, X69).

A locus property

The pKs that passes through A9, B9, C9 must have their pole on psK(X41, X9, X1) and their pivots on K1433.

For instance, pK(X220, X9) splits into the cevian lines of X(9), pK(X1, X144) = K202, pK(X2124, X7).

A pencil of cubics

K1433 and K977 pass through A, B, C, X(7), X(9) and four points on the axes of the Mandart ellipse, among them M1, M2.

K977 is the image of K003 under the symbolic substitution SS{a → √a} hence every cubic of the pencil generated by K1433 and K977 must be a cubic of the pencil generated by K003 and K443. This pencil also contains K019, K187, K376, K810, K851passing through A, B, C, X(3), X(4) and the four foci of the inconic with center X(3) among them X(46357), X(46358).