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K427

see the general equation in Table 33

X(4), X(30), X(265), X(382), X(1539)

A', B', C' : images of A, B, C under h(H, -1/2)

K427 is the cubic Lk with k = -1/2. See Table 33.

It is the only circular cubic of this type. The singular focus F is the midpoint of H X(265).

The real asymptote is the parallel to the Euler line at X(113) and meets the cubic at X(1539). It follows that any line through X(1539) meets the cubic again at P, Q and the midpoint M of PQ lies on the circle with center H, passing through X(1539). This has the same radius as the Euler circle.

K427 is the locus of pivots of circular pKs whose orthic line passes through X(381). See CL035.

The homothety h(H, -2) transforms K427 into another circular circumcubic passing through X(4), X(20), X(30), X(146), X(265), X(1138), X(1294).