In the realm of elliptic curves, the collinearity of three points is of a fundamental importance because this condition allows us to define on the curve a law of Abelian group, the study of which is the subject of endless enigmas and guesses, typical of a fascinating theory.This collinearity of three points, being a geometric concept is not exclusive to elliptic curves, of course.
On the other hand, it is well known that singular cubics (genus $0$) admit a rational parameterization with a rational parameter (non-singular cubics, genus $1$, admit also a rational parameterization but with a transcendental parameter, the function $\wp (z)$ of Weierstrass).
I want here post a question joining the concepts of parameter and collinearity. By instance, for the case of a cusp cubic defined by $$(x,y)=(t^2,t^3)$$ it is proved that $t_1t_2t_3=1$ is equivalent to the fact that the points $P_i=(t_i^2,t_i^3);\space i=1,2,3$ are collinear.
Let now $P_i(x_i,y_i);\space i=1,2,3$ be three points of the nodal cubic defined by $$(x,y)=\left(\frac{4t}{(1-t)^2},\frac{4t(1+t)}{(1-t)^3}\right)$$
Prove that $$P_1,P_2,P_3\text{ are collinear} \iff \frac {1}{t_1}+\frac {1}{t_2}+\frac {1}{t_3}= 0$$
