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K952

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X(4), X(13), X(14), X(30), X(477), X(5655), X(11586), X(15742)

X(11586) = reflection of X(13) in its trilinear polar

X(15743) = reflection of X(14) in its trilinear polar

see remark below

Geometric properties :

K952 is the orthopivotal cubic O(X30).

Its singular focus X(2) lies on the real asymptote, namely the Euler line, which is at the same time the orthic line since K952 is a circular K0+.

The polar conic of X(30) is the rectangular hyperbola (H) with center X(2) and asymptotes the Euler line and its perpendicular at X(2).

K952 belongs to the pencil of orthopivotal cubics with orthopivot on the Euler line that contains K001, K023, K059, K060, K313, K329, K479, K808. All these circular circum-cubics pass through X(4), X(13), X(14), X(30).

It also belongs to the pencil containing K003, K123, K315, K329. All these circum-cubics pass through X(4) and five other points lying on the Thomson-Jerabek hyperbola. See K002.

***

Remark : the locus of M such that X(30), M, the reflection of M in its trilinear polar is the Kiepert hyperbola.

More generally, if P is a fixed point not lying on the line at infinity, the locus of M such that P, M, the reflection of M in its trilinear polar is is a cubic called Thomson centroidal cubic in CL040.