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K1067

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X(6), X(110), X(512), X(1379), X(1380), X(2574), X(2575)

vertices of the cevian triangle of X(110)

vertices of the anticevian triangle of X(6) = tangential triangle

Geometric properties :

Under the mapping 𝚹 : P → aQ mentioned in the page K1065, the Brocard axis and K1067 are transformed into the same cubic (K), namely psK(X512 ÷ X1501, X670, X4), a nodal cubic with node X(76).

K1067 has three real asymptotes : one is perpendicular at X(6) to the Brocard axis and two are parallel at X(5181) to those of the Jerabek hyperbola whose complement is the bicevian conic (H) = C(X2, X110). (H) passes through X(3), X(5), X(6), X(113), X(141), X(206), X(942), X(960), X(1147), X(1209), X(1493), X(1511), X(2574), X(2575), X(2883) and also X(5181).

Note that the tangents to K1067 at A, B, C, X(110) are also perpendicular to the Brocard axis since the polar conic of X(512) is the circum-hyperbola passing through X(110), X(351), X(512), X(690), X(3124), etc.

Its perspector is : X(21906) = a^2 (b-c)^2 (b+c)^2 (2 a^2-b^2-c^2) : : and its center is : X(21905) = a^2 (b-c) (b+c) (2 a^2-b^2-c^2) (a^2 b^2+b^4+a^2 c^2-4 b^2 c^2+c^4) : : .

K1067 is the isogonal transform of K242, also the barycentric products X(6) x K242, X(110) x K237, X(1) x pK(X6, X662).

Its isotomic transform is pK(X1502, X670).

See also Table 34.