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X(2), X(97), X(251), X(324), X(8024), X(11143), X(11144), X(42008), X(46806), X(46807), X(46808), X(46809), X(51541) A', B', C' traces of the Brocard axis on the sidelines of ABC |
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Geometric properties : |
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K1285 is the isogonal transform of K1284 = cK(#X6, X1576). Its barycentric product by X(523) is the trident K1072 = cK(#X523, X6). K1285 is a crunodal isotomic isocubic with node X(2) and nodal tangents passing through X(3) (Euler line) and X(6). If P, Q are two points on the Brocard axis, inverse in the Brocard circle, then the lines GP and GQ meet the cubic again at two points P' and Q' which are isotomic conjugates. It follows that the line P'Q' envelopes the pivotal-conic (C) which is inscribed in the antimedial triangle (anticevian triangle of G), tangent at X(20), X(193) to the nodal tangents X(2)X(3), X(2)X(6) respectively. A fairly simple parametrization of K1285 is given by S(t) = (-t a^2+b^2+c^2) / (-a^2+t b^2+t c^2) : : , where t is a real number or ∞ or an expression of global degree 0 in a, b, c. Note that S(t) and S(1/t) are isotomic conjugates. K1285 meets the sidelines of ABC at three collinear points A', B', C' which lie on the trilinear polar of X(110), namely the Brocard axis. Its isotomic transform is the circum-conic with perspector X(850) passing through {76, 264, 276, 290, 300, 301, 308, 313, 327, 349, 1502, 2367, 3114, 3115}. Both curves have the same tangents at A, B, C. The contact-conic has perspector X(30476) and passes through X(i) for i in {253, 290, 683, 1975, 2996, 2998, etc}. Its isotomic transform is the line that passes through the points of inflexion of the cubic. This line is the polar line of G in (C). It contains X(i) for i in {20, 185, 193, 194, 511, 3164}. ***
S(1) = S(-1) = X(2). S(0) = X(8024) and S(∞) = X(251). Every other value of t is related to a group of four distinct points on K1285, namely S(t), S(1/t), S(-t), S(-1/t). The lines through S(t), S(1/t) and S(-t), S(-1/t) are tangent to (C). The lines through "opposite" points S(t), S(-t) and S(1/t), S(-1/t) concur at Y on the cubic. Y is the barycentric quotient X(6656) ÷ X(7770). The lines through "inverse-opposite" points S(t), S(-1/t) and S(-t), S(1/t) concur at the isotomic conjugate Y' of Y also on the cubic. The lines GS(t) and GS(-t) meet the Brocard axis at two points, inverse in the circle with diameter X(32)X(39) which is orthogonal to the Brocard circle. ***
Let M = u:v:w ≠ G be a point. The parametrization given by S(t) = (-t u+v+w) / (-u+t v+t w) : : is that of the cubic K(M) = cK(#G, R), where R = u/(v-w) : : , is the trilinear pole of the line passing through M and G/M, the G-Ceva conjugate of M. Hence, K(M) = K(G/M). Obviously K(X3) = K(X6) = K1285. K(M) is a nodal cubic with node G. The nodal tangents pass through M and G/M. The pivotal-conic is inscribed in the antimedial triangle. It is tangent to the nodal tangents at their intersections with the anticomplement of the anticomplement of the line through M and G/M, equivalently its homothetic under h(G, 4). The contact-conic has perspector the anticomplement of the anticomplement of the isotomic conjugate of R. Its isotomic transform is the line passing through the points of inflexion of K(M). S(0) = v w (v+w) : : is the tM-Ceva conjugate of cM, where tM and cM are the isotomic conjugate and the complement of M. S(∞) = u (u+v)(u+w) : : is the isotomic conjugate of S(0). |