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X(4), X(15), X(16), X(511), X(1113), X(1114), X(3164), X(5167)

U15 = X(15)X(3164) ∩ X(16)X(5167)

U16 = X(16)X(3164) ∩ X(15)X(5167)

isogonal conjugates of the infinite points of the Steiner ellipses

other points below

Geometric properties :

K1445 is a focal cubic with singular focus X(4).

It is the locus of contacts of tangents drawn through X(4) to the circles passing through X(15) and X(16), or, equivalently, the locus of the common points of a circle with the polar line of X(4) in this circle. This line passes through X(3164), the finite point on the real asymptote, the anticomplement of the isotomic conjugate of O.

It is also the locus of foci of conics inscribed in the triangle X(4)X(15)X(16) which are centered on the Brocard axis. Hence, K1445 is an isogonal nK in this triangle and two isogonal conjugates share the same tangential.

It follows that K1445 passes through the isogonal conjugates (in this triangle) T3, T4 of X(1113), X(1114) and also :

Y3 = X(15)X(1113) ∩ X(16)T3 and Z3 = X(16)X(1113) ∩ X(15)T3

Y4 = X(15)X(1114) ∩ X(16)T4 and Z4 = X(16)X(1114) ∩ X(15)T4

Note that {U15, U16}, {Y3, Z3} and {Y4, Z4} are pairs of isogonal conjugates.

A construction

The perpendicular at H to a variable line (L) passing through X(3164) meets the Lemoine axis at Ω. The circle with center Ω passing through X(15) and X(16) meets (L) at two points of K1445.