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X(2), X(13), X(14), X(125), X(543), X(1316), X(5465), X(9169)

A2, B2, C2 : vertices of the second Brocard triangle

K509 is the locus of X(110) of all Kiepert triangles when the base angle phi varies. See Table 32.

K509 is also the Psi transform of the Kiepert hyperbola where Psi is the involution met in "Orthocorrespondence and Orthopivotal Cubics" §5. See a generalization below and the analogous strophoids K794, K795, K796.

The node is the centroid G with nodal tangents the axes of the Steiner ellipses. The singular focus F is X(9169) = Psi(X671), X671 being the antipode of G on the Kiepert hyperbola.

For any point M on K509, the perpendicular bisector (L) of GM envelopes the parabola (P) with focus F and directrix the line GX(99).

The contact N of (L) and (P) is the center of a circle passing through G and tangent at M to K509.

(P) is obviously tangent to the axes of the Steiner ellipses.

Now, if we take two antipodes m and m' on the Kiepert hyperbola and their Psi-images M and M' on K509, the line MM' envelopes another parabola with same directrix and with focus the reflection of G in F.

This latter point is Psi(X115) = X(6792).

Generalization

The Psi transform of the circum-conic (C) with perspector P is generally a bicircular quartic (Q) passing through the vertices of the second Brocard triangle and the centroid G which is a singular point.

• if (C) is the circum-circle of ABC hence P = X(6), (Q) splits into the Brocard circle and the circle with center G, radius 0.

• if (C) is a parabola (i.e. P lies on the Steiner inellipse), G is a cusp.

• if (C) is a rectangular hyperbola (i.e. P lies on the orthic axis), the nodal tangents at G are perpendicular.

• if (C) passes through G (i.e. P lies on the line at infinity), (Q) splits into the line at infinity and a circular nodal cubic (K) with node G and singular focus on the McCay circumcircle. See ETC, X(7606).

K509 is obtained with P = X(523) verifying the last two conditions above.

When P = X(512), the conic (C) passes through X(2), X(6), etc, and (K) is K793.