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∑ SA x^2 (c y - b z) = 0

X(1, X(4), X(46), X(7040)

feet of the altitudes

X40-OAP points, see K826 and Table 53

other points M1, M2 and details below

K691 and K692 are two members of a pencil of cubics which contains the cubic decomposed into the line (L) = X(4)X(9) and the circum-conic (C) passing through X(1), X(2).

K691 and K692 are the only pKs of this pencil whose base-points are A, B, C, X(1), the intersections M1, M2 of (L) and (C) and three points at infinity which are those of the Simson lines passing through X(1). See the papers Asymptotic Directions of Pivotal Isocubics and The Cevian Simson Transformation.

There is one and only one K+ in the pencil with asymptotes concuring at X(238).

Note that M1, M2 are X(19)-isoconjugate points.

K691 = pK(X19, X4) is tangent at A, B, C to the internal bisectors and the polar conic of X(1) is the Feuerbach hyperbola.

K691 meets the circumcircle at the same points as pK(X6, X3868).

Its isogonal transform is pK(X48, X1) and its isotomic transform is pK(X304, X75).