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X(12248) = reflection H' of H about X(104) 

A circumconic with perspector Ω on the orthic axis is a rectangular hyperbola (H) with center G/Ω on the nine points circle and meeting the circumcircle again at the reflection Q of H about G/Ω. The circle with center Q passing through H meets this hyperbola at H and three other points which are the vertices of an equilateral triangle. The pivotal cubic pK with pole Ω on the orthic axis and with pivot the reflection H' of H about Q is an equilateral pK (a pK60) passing through the vertices of this equilateral triangle. When Ω = X(650), (H) is the Feuerbach hyperbola and the pK60 is K542 = pK(X650, X12248). 
