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X(2), X(182), X(524), X(3413), X(3414), X(6671), X(6672), X(10007), X(10008), X(10011), X(11132), X(11133), X(42838), X(42840), X(51373), X(57575), X(57576)

X(64939) → X(64949)

other points below

Geometric properties :

Let (L) be the trilinear polar of X(99), which passes through X(2), X(6) and many other points.

For P on (L), the G-Ceva conjugate Q of P lies on the bicevian conic C(X2, X99), a rectangular hyperbola homothetic to the Kiepert hyperbola (K), with center X(620), passing through X(2), X(3), X(39), X(114), X(618), etc.

When P traverses (L), the midpoint M of P, Q lies on K1364.

K1364 is a cuspidal cubic with cusp G and cuspidal tangent passing through X(99).

K1364 has three real asymptotes : one is the line {524,620) and two are parallel at X(6722) to those of the Kiepert hyperbola. X(6722) is the complement of X(620).

K1364 meets the sidelines of ABC at three points U, V, W on the line {99,110) and six points on an ellipse (E) homothetic to the Steiner ellipse under h(X99, 3/4), with center X(620), passing through X(99), X(115). Note that (E) is tangent to the Steiner circum-ellipse and to the Steiner in-ellipse at X(99) and X(115) respectively.

See a generalization in K869. See also K934, a very similar cubic.

The barycentric product of X(115) and K1364 is the cuspidal cubic K1381.