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midpoints of ABC

cevians of X(69) = isotomic conjugate of H.

A P-pedal cubic is the locus of point M whose pedal circle passes through P. See CL036. This cubic is a K+ if and only if P lies on Q032, the point pedal quartic.

Q032 is a bicircular quartic. It meets the sideline BC at the midpoint Ma of BC, the cevian of X(69) and two other points lying on the circle centered at Ma whose radius is sqrt(b^2+c^2)/2, the quadratic mean of the sidelengths AB, AC.