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too complicated to be written here. Click on the link to download a text file. |
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X(514), X(1019), X(1577), X(23594), X(23595), X(23596), X(23597), X(23598), X(23599) |
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Geometric properties : |
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K1074 is a conico-pivotal nodal cubic with pivotal conic the parabola with focus X(1054) and directrix the line passing through X(1), X(2). K1074 is a trident having an asymptote which is the trilinear polar L(X7) of the Gergonne point and also a parabolic asymptote (P) with equation : ∑ sb sc x^2 + [a sa + (b - c)^2] y z = 0, where sa = (– a + b + c)/2 as usual. L(X7) and (P) meet at X(7658). The tangent to (P) at this point is L(X9311) which contains the three finite points of inflexion of K1074 but one only is real.
1. K1074 is the image under 𝛕 𝛕 2. Peter Moses observes that K1074 is the locus of points Z(t) = (b-c) (t a + b + c) (t a + t b + c) (t a + t c + b) : : , where t is a real parameter or infinity. For example Z(0) = X(1577), Z(1) = X(514), Z(∞) = X(1019). Note that Z(t) and Z(1/t) are swapped under the isoconjugation with fixed point X(514) hence the line Z(t)Z(1/t) is tangent at T(t) to the pivotal parabola. The harmonic conjugate of T(t) in Z(t), Z(1/t) is the third point of K1074 on this line Z(t)Z(1/t). |