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X(6), X(74), X(511), X(1495), X(1976), X(13414), X(13415)

See the notations and the general properties at CL030.

The isogonal transform of K223 is K953 = nK0(X2, X525) = cK(#X2, X525). K223 is also the SS{a -> a^2} image of K040.

(Comments below by Wilson Stothers)

C(R) is the Jerabek Hyperbola. R* = X(112). T(R*) is the line X(6)X(25).

K223 is :

1. The locus of X such that XX* has mid-point on the Brocard Axis.
2. The locus of X such that XX* is divided harmonically by the Lemoine Axis (T(K)) and the tangent to C(R) at K - this is the line KX(25) and the tripolar of R*.
3. The K-Hirst inverse of C(R).

K223 contains :

• K, the node,
• X(74) as the intersection of C(K) and C(R),
• X(511) as the infinite point of the Brocard axis or as the isoconjugate of O,
• X(1495) as the intersection of T(K) and T(R*),
• X(1976) as the isoconjugate of X(511).
• S1, S2 as in Table 62. K223 is actually also invariant under the JS involution described in Table 62.

The nodal tangents are

• The tangents from K to I(R*)
• The tripolars of the intersections of T(R*) and C(K)
• The lines from K to the intersections of C(R) and the Lemoine axis.

The pivotal conic touches the Lemoine axis at R. This identifies it as an inconic of the tangential triangle with center X(924).