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X(3), X(4), X(1141), X(1154), X(2070), X(25043), X(25044), X(33565), X(38896), X(38897)

imaginary foci of the MacBeath inconic (M)

M1, M2 see below

Geometric properties :

K1154 is a focal isogonal nK with singular focus X(1141). Also, K1154 is spK(X1154, X5) as in CL055.

Its orthic line is the Euler line of the orthic triangle passing through X(5), X(51), X(52), etc, whose infinite point is X(1154) on the cubic. The two remaining common points M1, M2 are isogonal conjugates hence they lie on the circum-conic passing through X(54), X(95), X(96), X(252), X(1141), X(1166), etc, and obviously on the Napoleon cubic K005.

K1154 is the locus of

• foci of inconics with center on the orthic line above, hence tangent to the trilinear polar of the root X(18314).

• contacts of tangents drawn through X(1141) to the circles passing through M1, M2. One of them is the polar conic of X(1141), passing through X(476).

• point M such that the directed angles (MX3, MX1154) and (MX1141, MX4) are equal (mod. π).

• pivots of circular pKs whose orthic line passes through X(54). See CL035.