The x-coordinates of those three tangent points lie on a line y=d, where d is a y-coordinate of the inflection point of the antiderivative of the quartic polynomial.
Moreover, the three points are separated evenly; there x-coordinates have an arithmetic progression.
There uniquely exists one tangent line that is tangent to a quartic polynomial at two distinct points.
There also exists a singe line that is tangent to the aforementioned quartic polynomial, and parallel to the aforementioned unique tangent line.