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too complicated to be written here. Click on the link to download a text file. |
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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 |
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Geometric properties : |
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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.
• 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. |