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X(2), X(5), X(6), X(13), X(14), X(15), X(16), X(3070), X(3071), X(5103)

X(42562) → X(42575)

For any point M on the Brocard axis, the isogonal conjugate M* of M lies on the Kiepert hyperbola and the line MM* envelopes the Brocard-Kiepert quartic Q073.

This line MM* meets the Kiepert hyperbola again at N*. The tangents at M* and N* to the Kiepert hyperbola meet at P and, when M traverses the Brocard axis, the locus of P is the Brocard-Kiepert cubic K458.

K458 is an unicursal cubic with an isolated point X(5) i.e. an acnodal cubic. It has three real inflexion points which are the Lemoine point K and the isodynamic points X(15), X(16). The tangents at X(15), X(16) meet at X = X(3628) on the Euler line such that OX = 3/4 OX(5) or GX = 1/4 GX(5).

The isogonal conjugates G, X(13), X(14) of these inflexion points are the contacts of the cubic with the Kiepert hyperbola. In other words, K458 is tritangent to the Kiepert hyperbola at these points.

K458 meets the Evans conic at six identified points namely X(13), X(14), X(15), X(16), X(3070), X(3071). The tangents at X(3070), X(3071) – two points on the line HK – pass through G.

***

K458 is a KHO-curve, see K1191 and CL075 for a generalization. See also K1225, a similar acnodal cubic.

Its KHO-equation is : x^2 (2y - 3z) - 3 (2y - z) (y - z)^2 = 0.

Its Hessian H458 is : x^2 (10y - 9z) - 3 (2y - 3z) (y - z)^2 = 0 and its only undecomposed preHessian P458 is : x^2 (2y - z) + (2y + z) (y - z)^2 = 0.

These three cubics obviouly share the same singularity X(5) and the same points of inflexion X(6), X(15), X(16).

H458 also contains X(3091) and two points R1, R2 with KHO-coordinates (±3, 6, 7), now X(42581), X(42580) in ETC.

The polar conic of X(3091) decomposes into the two lines X(5)X(6) and X(6)X(3090).

R1 lies on the lines {X5,X14}, {X13,X1656}, {X15,X3091}, {X62,X3090}.

R2 lies on the lines{X5,X13}, {X14,X1656}, {X16,X3091}, {X61,X3090}.

X(42582), X(42583) also lie on H458.

***

P458 also contains X(20)and defines a conjugation on K458, mapping M = u:v:w onto N with KHO-coordinates :

3(v - w)^2 - u(v - 3w) : (u - 2v + w) (u + v + w) : u(2u + w)-(2v - w) (3v - w).

N is the center of the (decomposed) polar conic of M with respect to P458.

The points Q1, Q2, Q3, Q4 on K458 are those with KHO-coordinates :

Q1 = (12-7 Sqrt[3], 5-3 Sqrt[3], 3-2 Sqrt[3]), SEARCH = 2.56458915404377, on the lines {X13,X3070}, {X15,X3071}.

Q2 = (12+7 Sqrt[3], 5+3 Sqrt[3], 3+2 Sqrt[3]), SEARCH = 0.472340128000023, on the lines {X13,X3071}, {X15,X3070}.

Q3 = (-12-7 Sqrt[3], 5+3 Sqrt[3], 3+2 Sqrt[3]), SEARCH = 2.19576248117195, on the lines {X14,X3070}, {X16,X3071}.

Q4 = (-12+7 Sqrt[3], 5-3 Sqrt[3], 3-2 Sqrt[3]), SEARCH = 4.10270357723251, on the lines {X14,X3071}, {X16,X3070}.

These points are X(42562), X(42563), X(42564), X(42565) in ETC.