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X(3), X(24), X(52), X(54), X(2917), X(3432), X7488), X(8823), X(8824), X(8907) P and P*, see below 

The Kosnita triangle is formed by the circumcenters Oa, Ob, Oc of the three triangles OBC, OCA, OAB. It is perspective to ABC at X(54), the Kosnita point and homothetic to the tangential triangle under h(O,1/2). Its circumcenter is X(1658), the midpoint of O X(26). Its orthocenter is X(1147). More informations in Darij Grinberg's FG paper. K388 is a pivotal cubic passing through the vertices of the Kosnita triangle. Its pole is X(571) (on the Brocard axis) and its pivot is P = X(7488), the midpoint of O X(2937), the intersection of the Euler line and the line X(52)X(54). Its isopivot P* is the intersection of the line X(3)X(2917) and the tangent P X(311) at P to the cubic. K388 is tangent at O to the Brocard axis. K388 also contains :
The tangents at X(52), Oa, Ob, Oc concur. It follows that K288 is also a pK with respect to the Kosnita triangle. See also Table 31 : Kosnita curves. The cevian – resp. anticevian – triangle of M is perspective to the Kosnita triangle if and only if M lies on pK(X2, X311) – resp. pK(X571, X1994) – and in both cases, the locus of the perspector is K388. 
