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infinite points of the line BC and the perpendicular bisector of BC

The in/excenters of the triangles with vertices B, C and a variable point M on the parallel at A to the line BC lie on the cubic K456A. The two other cubics K456B and K456C are defined similarly. See the figure below.


K456A is an axial cubic symmetric about the perpendicular bisector La of BC. It has a real asymptote passing through the midpoints of AB and AC. It has a parabola asymptote with axis La. The cubic and the parabola have a sextactic point in common at infinity, that of La.

Obviously, K456A contains the reflections of the in/excenters of ABC in La and also the in/excenters of the isosceles triangle BCA'.