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X(6), X(10), X(19), X(46), X(55), X(65), X(209), X(1824)

vertices of the orthic and extangents triangles

K391 meets the sidelines at the vertices of the orthic triangle and has its tangents at A, B, C concurrent at X(42). These are necessary but not sufficient conditions for the cubic to be a pK. Indeed, K391 is not a pK since H does not lie on the curve.

The third intersection of the cubic with orthic A - sideline is A' = a^2 (b + c) : b (c^2 - a^2 + b c) : c (b^2 - a^2 + b c).

The third intersection of the cubic with extangents A - sideline is A" = -(b + c)^2 : b^2 : c^2 on the symmedian AK.

A"B"C" is perspective to ABC and tangential triangle at K, orthic at X(1824), medial at X(10) and extouch at X(14973) = X(37), X(42) /\ X(181), X(594). Recall that the orthic and extangents triangles are homothetic at X(19).

K391 has the same asymptotic directions as K175 = pK(X32, X1).

It meets the circumcircle at the same points as the pK with pivot X(10), poleX(14974) = a^2(a b + a c - b c - SA) : : , and also the pK with pivot X(19), pole X(14975) = a^3 SB SC (2 SA + b c): : .

K391 is also psK(X2333, X4, X6) in Pseudo-Pivotal Cubics and Poristic Triangles.