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K1124

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X(6), X(154), X(1350), X(8667), X(33928)

infinite points of the symmedians

vertices of these triangles :

• Thomson (points Qi)

• tangential of Thomson (points Ri)

other points below

Geometric properties :

K1124 is related to Table 71. A cubic K(T) meets the Euler line at X(3) counted twice and another point E(T). The line passing through X(6) and E(T) meets K(T) again at E'(T). When T varies, the locus of E'(T) is K1124.

Hence every cubic K(T) meets K1124 at eight fixed points (Q1, Q2, Q3, R1, R2, R3, O counted twice) and then E'(T) is their last common point.

K1124 is a K+ with three real asymptotes parallel at X(5085) to the symmedians of ABC.

K1124 is a nodal cubic with node X(6) and two always real nodal tangents.

K1124 meets the circumcircle (O) at the vertices Q1, Q2, Q3 of the Thomson triangle and three other points Z1, Z2, Z3. These points lie on a family of nK0(Ω, R) with Ω on the line through X(32), X(184) and R on the Brocard axis.