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X(2), X(3), X(6), X(10), X(37), X(65), X(429), X(960), X(1193), X(1211), X(2051), X(2092), X(2092) midpoints of ABC 

K253 is the locus of poles of all pK which have the same asymptotic directions as pK(X6, X81) and pK(X2, X314) = K254. It meets the circumcircle at the same points as pK(X6, X1211) and the Steiner ellipse at the same points as pK(X2, X2092). See "Asymptotic Directions of Pivotal Isocubics" in the downloads page. 
