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X(7), X(11), X(13), X(14), X(80), X(528), X(13576), X(17777)

other points below

Geometric properties :

K1051 is the orthopivotal cubic O(X11), a nodal cubic with node X(80). See Q015.

Its singular focus is X(18343), a point on the Fuhrmann circle (F).

K1051 meets the Feuerbach hyperbola (H) at A, B, C, X(7) and X(80) twice.

K1051 meets the circumcircle (O) at A, B, C, the circular points at infinity and S, on the lines {11, 110}, {80, 101}, {109, 2006}, etc, with SEARCH = 6.18099242390188.

K1051 meets the Fermat axis at X(13), X(14) and T, also on the line {11, 110} with SEARCH = 3.35961729909503.

The real asymptote is the line passing through X(528), X(644) hence parallel to the lines {2, 11}, {7, 664}, {9, 80}, etc.

K1052 is the isogonal transform of the K1051.

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Points on the sidelines of ABC and a construction

Let Ha be the pedal of the centroid G on the altitude AH. The intersection U of BC and the line X(11)Ha lies on K1051. V and W are defined similarly. Note that Ha, Hb, Hc lie on the orthocentroidal circle.

Let L be the tangent at X(80) to the Feuerbach hyperbola (H). L is parallel to the line X(4)X(8) and passes through X(517).

The trilinear polar of a variable point on L meets the lines UX(80), VX(80), WX(80) at three points which are the vertices of a triangle perspective to ABC. The perspector lies on K1051.

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Related parabolas

Let (P1) be the parabola with focus X(18343) and directrix the line passing through X(1), X(528) hence parallel to the real asymptote of K1051. The envelope of circles centered on (P1) and passing through X(80) is K1051. (P1) is called the deferent parabola of K1051.

Let (P2) be the parabola homothetic of (P1) under h(X80,2), with directrix the line X(528), X(3243) and focus the reflection of X(80) in X(18343). The pedal curve of (P2) with respect to X(80) is K1051.