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K543

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X(5667) = reflection of H about X(107)

A circum-conic 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.

This pK60 has three real concurring asymptotes at X if and only if Ω = X(1990) and this pK60+ is K543 = pK(X1990, X5667). X = X(14847).