If six points of an elliptic curve are contained in a conic, then their sum is $O$. Let $C$ be a projective cubic without singular points and $O\in C$ an inflexion point. We consider the addition in $C$ with $O$ as neutral element. If $R_{1},...,R_{6}\in C$ are different points such that there exists a conic that contains all of them, then $R_{1}+...+R_{6}=O$. I think I have to use Pappus/Pascal's theorem (depending on the irreducibility of the conic) and the next result: $a,b,c\in C$ lie on the same line if and only if $a+b+c=O$. The thing is that I do not know how to combine them to solve the exercise. Observation: This exercise is a particular case of theorem 9.2. of chapter 6 in Walker's algebraic curves book, but I would like to use more elementary tools.

Let $R_1, \ldots, R_6$ be distinct points on $C$. Choose three pairs of them, form the corresponding straight lines and call $C'$ the reducible cubic consisting of such lines. Consider the linear system of cubics passing through the 9 points in the intersection of $C$ with $C'$. Then, if $R_1, \ldots, R_6$ lie on a conic $K$, we have a reducible cubic consisting of $K$ and a straight line $r$ passing through the remaining 3 points in the intersection. The sum of all 9 points is 0 and the sum of the three points in $r$ is 0, hence the thesis follows.