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X(2), X(3), X(376), X(46949), X(46950)

infinite points of K243

vertices of the Thomson triangle Q1Q2Q3

other points below

Geometric properties :

K1260 is K443 for the Thomson triangle Q1Q2Q3. Recall that K443 is psK(X3, X69, X3) and spK(X4, X3) in ABC.

K1260 is a central cubic with center O, point of inflexion with tangent passing through X(1495).

K1260 is a member of the pencil is generated by K758 and the decomposed cubic which is the union of the Euler line and the circumcircle. This pencil also contains K764, K1261.

Points on K1260 :

• vertices Q1, Q2, Q3 of the Thomson triangle with tangents concurring at X(5646), the Lemoine point of Q1Q2Q3.

• reflections R1, R2, R3 of Q1, Q2, Q3 in O.

• foci of (C), conic with center O inscribed in Q1Q2Q3 and R1R2R3. F1 and F2 are the real foci.

• contacts T1, T2, T3 of (C) with the sidelines of Q1Q2Q3. T1T2T3 is the cevian triangle of X(69) for the Thomson triangle – a point on the line {6,376} – hence K1260 is a psK for this triangle.

• contacts S1, S2, S3 of (C) with the sidelines of R1R2R3.

• infinite points of K243. The six finite remaining common points lie on the rectangular hyperbola (H) passing through X(3), X(6), X(376), X(1992), X(2574), X(2575).

• the tangential of G lies on the lines {2,6}, {111,1350} and the tangential of X(376) lies on the lines {6,1296}, {376,524,1350}. These points are :

a^2 (a^4-10 a^2 b^2+b^4-10 a^2 c^2+26 b^2 c^2+c^4) : : , SEARCH = -68.5645374259164,

a^2 (a^8-8 a^6 b^2-22 a^4 b^4+32 a^2 b^6-3 b^8-8 a^6 c^2+128 a^4 b^2 c^2-112 a^2 b^4 c^2-8 b^6 c^2-22 a^4 c^4-112 a^2 b^2 c^4+86 b^4 c^4+32 a^2 c^6-8 b^2 c^6-3 c^8) : : , SEARCH = 82.1292632143091.

They are now X(46949) and X(46950) in ETC.