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K476

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X(4), X(67), X(523), X(879), X(935)

K476 is a circular cubic related with CL051.

The singular focus is X(1352) midpoint of X(4)-X(69) and reflection of K about X(5), the real asymptote is X(6)-X(523), the orthic line is X(5)-X(523).

K476 is a nK with pole and root the barycentric product X(8791) of X(4) and X(67). It meets the sidelines of ABC on the trilinear polar of this root namely the line X(512)-X(1843).

The polar conic of X(879) meets K476 again at four identified points X(4), X(67), X(523), X(935) hence the tangents at these four points pass through X(879).

The vertices of the diagonal triangle of these four points also lie on the cubic which is therefore a pK in this triangle with pivot X(879).

A property related with the 4th Brocard triangle A4B4C4

Let U, V, W be the traces on the sidelines of ABC of the trilinear polar of P. The lines UA4, VB4, WC4 concur (at Q) if and only if P lies on K476. Le locus of Q is nK(X25, X468, X4) passing through X(4), X(6), X(112), X(523). The corresponding pairs {P, Q} are {X(i), X(j)} for {i,j} = {4,4}, {67,523}, {523,6}, {879, 30 x 98}, {935,112}.

In particular, A4, B4, C4 lie on the lines passing through the orthocenter H and the traces of the orthic axis.