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X(1), X(4), X(5), X(169), X(3491)

excenters

midpoints of the altitudes

infinite points of K006

projections of X(4) on the six bisectors

K858 = Cax(X4) is the locus of centers of inscribed conics having an axis passing through X(4). See CL047 for the general case and also K258 = Cax(X3).

K858 is a nodal cubic with node X(4), the nodal tangents being parallel to the asymptotes of the rectangular circum-hyperbola passing through X(20).

K858 is also the image of the line at infinity under the involution 𝝋 mentioned in the page K856. This gives the following construction. For M on the line at infinity, let (H) be the diagonal rectangular hyperbola passing through M and the in/excenters and (L) the line MX(4). (L) and (H) meet at M and another point N on K858.