What a Beautiful Curve!

Consider the curve defined by the parametric equation: $\begin{cases} x=\frac { t }{ 1+{ t }^{ 3 } } \\ y=\frac { { t }^{ 2 } }{ 1+{ t }^{ 3 } } \end{cases},\quad t\neq -1$

Suppose that 3 tangents are drawn at 3 distinct points on the curve, and the 3 points are collinear.

Now is where things come crazy

Given that each tangent intersects the curve once again at a point other than the tangent point, would these points of intersection be collinear also?

Big Hint: you may use the fact that $P({ t }_{ 1 })$ , $P({ t }_{ 2 })$ , $P({ t }_{ 3 })$ are collinear iff ${ t }_{ 1 }{ t }_{ 2 }{ t }_{ 3 }=-1$ , where $P( t )$ denotes any points on the curve with parameter $t$ .

Extra challenge: Prove the Hint please

If the answer is sometimes, please include the specific condition(s) in the solution